Complex Analysis, Third Edition (Undergraduate Texts in Mathematics)

Undergraduate Texts in Mathematics Editorial Board S. Axler K.A. Ribet For other titles Published in this series, go t...

Author: Joseph Bak | Donald J. Newman

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Undergraduate Texts in Mathematics Editorial Board S. Axler K.A. Ribet

For other titles Published in this series, go to http://www.springer.com/series/666

Joseph Bak • Donald J. Newman

Complex Analysis Third Edition

1C

Joseph Bak City College of New York Department of Mathematics 138th St. & Convent Ave. New York, New York 10031 USA [email protected]

Editorial Board: S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA [email protected]

Donald J. Newman (1930–2007)

K. A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720 USA [email protected]

ISSN 0172-6056 ISBN 978-1-4419-7287-3 e-ISBN 978-1-4419-7288-0 DOI 10.1007/978-1-4419-7288-0 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010932037 Mathematics Subject Classification (2010): 30-xx, 30-01, 30Exx © Springer Science+Business Media, LLC 1991, 1997, 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface to the Third Edition

Beginning with the first edition of Complex Analysis, we have attempted to present the classical and beautiful theory of complex variables in the clearest and most intuitive form possible. The changes in this edition, which include additions to ten of the nineteen chapters, are intended to provide the additional insights that can be obtained by seeing a little more of the “big picture”. This includes additional related results and occasional generalizations that place the results in a slightly broader context. The Fundamental Theorem of Algebra is enhanced by three related results. Section 1.3 offers a detailed look at the solution of the cubic equation and its role in the acceptance of complex numbers. While there is no formula for determining the roots of a general polynomial, we added a section on Newton’s Method, a numerical technique for approximating the zeroes of any polynomial. And the Gauss-Lucas Theorem provides an insight into the location of the zeroes of a polynomial and those of its derivative. A series of new results relate to the mapping properties of analytic functions. A revised proof of Theorem 6.15 leads naturally to a discussion of the connection between critical points and saddle points in the complex plane. The proof of the Schwarz Reflection Principle has been expanded to include reflection across analytic arcs, which plays a key role in a new section (14.3) on the mapping properties of analytic functions on closed domains. And our treatment of special mappings has been enhanced by the inclusion of Schwarz-Christoffel transformations. A single interesting application to number theory in the earlier editions has been expanded into a new section (19.4) which includes four examples from additive number theory, all united in their use of generating functions. Perhaps the most significant changes in this edition revolve around the proof of the prime number theorem. There are two new sections (17.3 and 18.2) on Dirichlet series. With that background, a pivotal result on the Zeta function (18.10), which seemed to “come out of the blue”, is now seen in the context of the analytic continuation of Dirichlet series. Finally the actual proof of the prime number theorem has been considerably revised. The original independent proofs by Hadamard and de la Vallée Poussin were both long and intricate. Donald Newman’s 1980 article

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presented a dramatically simplified approach. Still the proof relied on several nontrivial number-theoretic results, due to Chebychev, which formed a separate appendix in the earlier editions. Over the years, further refinements of Newman’s approach have been offered, the most recent of which is the award-winning 1997 article by Zagier. We followed Zagier’s approach, thereby eliminating the need for a separate appendix, as the proof relies now on only one relatively straightforward result due of Chebychev. The first edition contained no solutions to the exercises. In the second edition, responding to many requests, we included solutions to all exercises. This edition contains 66 new exercises, so that there are now a total of 300 exercises. Once again, in response to instructors’ requests, while solutions are given for the majority of the problems, each chapter contains at least a few for which the solutions are not included. These are denoted with an asterisk. Although Donald Newman passed away in 2007, most of the changes in this edition were anticipated by him and carry his imprimatur. I can only hope that all of the changes and additions approach the high standard he set for presenting mathematics in a lively and “simple” manner. In an earlier edition of this text, it was my pleasure to thank my former student, Pisheng Ding, for his careful work in reviewing the exercises. In this edition, it as an even greater pleasure to acknowledge his contribution to many of the new results, especially those relating to the mapping properties of analytic functions on closed domains. This edition also benefited from the input of a new generation of students at City College, especially Maxwell Musser, Matthew Smedberg, and Edger Sterjo. Finally, it is a pleasure to acknowledge the careful work and infinite patience of Elizabeth Loew and the entire editorial staff at Springer. Joseph Bak City College of NY April 2010

Preface to the Second Edition

One of our goals in writing this book has been to present the theory of analytic functions with as little dependence as possible on advanced concepts from topology and several-variable calculus. This was done not only to make the book more accessible to a student in the early stages of his/her mathematical studies, but also to highlight the authentic complex-variable methods and arguments as opposed to those of other mathematical areas. The minimum amount of background material required is presented, along with an introduction to complex numbers and functions, in Chapter 1. Chapter 2 offers a somewhat novel, yet highly intuitive, definition of analyticity as it applies specifically to polynomials. This definition is related, in Chapter 3, to the Cauchy-Riemann equations and the concept of differentiability. In Chapters 4 and 5, the reader is introduced to a sequence of theorems on entire functions, which are later developed in greater generality in Chapters 6–8. This two-step approach, it is hoped, will enable the student to follow the sequence of arguments more easily. Chapter 5 also contains several results which pertain exclusively to entire functions. The key result of Chapters 9 and 10 is the famous Residue Theorem, which is followed by many standard and some not-so-standard applications in Chapters 11 and 12. Chapter 13 introduces conformal mapping, which is interesting in its own right and also necessary for a proper appreciation of the subsequent three chapters. Hydrodynamics is studied in Chapter 14 as a bridge between Chapter 13 and the Riemann Mapping Theorem. On the one hand, it serves as a nice application of the theory developed in the previous chapters, specifically in Chapter 13. On the other hand, it offers a physical insight into both the statement and the proof of the Riemann Mapping Theorem. In Chapter 15, we use “mapping” methods to generalize some earlier results. Chapter 16 deals with the properties of harmonic functions and the related theory of heat conduction. A second goal of this book is to give the student a feeling for the wide applicability of complex-variable techniques even to questions which initially do not seem to belong to the complex domain. Thus, we try to impart some of the enthusiasm

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apparent in the famous statement of Hadamard that "the shortest route between two truths in the real domain passes through the complex domain." The physical applications of Chapters 14 and 16 are good examples of this, as are the results of Chapter 11. The material in the last three chapters is designed to offer an even greater appreciation of the breadth of possible applications. Chapter 17 deals with the different forms an analytic function may take. This leads directly to the Gamma and Zeta functions discussed in Chapter 18. Finally, in Chapter 19, a potpourri of problems–again, some classical and some novel–is presented and studied with the techniques of complex analysis. The material in the book is most easily divided into two parts: a first course covering the materials of Chapters 1–11 (perhaps including parts of Chapter 13), and a second course dealing with the later material. Alternatively, one seeking to cover the physical applications of Chapters 14 and 16 in a one-semester course could omit some of the more theoretical aspects of Chapters 8, 12, 14, and 15, and include them, with the later material, in a second-semester course. The authors express their thanks to the many colleagues and students whose comments were incorporated into this second edition. Special appreciation is due to Mr. Pi-Sheng Ding for his thorough review of the exercises and their solutions. We are also indebted to the staff of Springer-Verlag Inc. for their careful and patient work in bringing the manuscript to its present form. Joseph Bak Donald J. Newmann

Contents

Preface to the Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1

The Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The Field of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The Solution of the Cubic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Topological Aspects of the Complex Plane . . . . . . . . . . . . . . . . . . . . . 12 1.5 Stereographic Projection; The Point at Infinity . . . . . . . . . . . . . . . . . . 16 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2

Functions of the Complex Variable z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Analytic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Differentiability and Uniqueness of Power Series . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 21 25 28 32

3

Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Analyticity and the Cauchy-Riemann Equations . . . . . . . . . . . . . . . . . 3.2 The Functions e z , sin z, cos z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 40 41

4

Line Integrals and Entire Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Properties of the Line Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Closed Curve Theorem for Entire Functions . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 45 52 56

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5

Contents

Properties of Entire Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Cauchy Integral Formula and Taylor Expansion for Entire Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Liouville Theorems and the Fundamental Theorem of Algebra; The Gauss-Lucas Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Newton’s Method and Its Application to Polynomial Equations . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Properties of Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Power Series Representation for Functions Analytic in a Disc . . 6.2 Analytic in an Arbitrary Open Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Uniqueness, Mean-Value, and Maximum-Modulus Theorems; Critical Points and Saddle Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 77 81

Further Properties of Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The Open Mapping Theorem; Schwarz’ Lemma . . . . . . . . . . . . . . . . . 7.2 The Converse of Cauchy’s Theorem: Morera’s Theorem; The Schwarz Reflection Principle and Analytic Arcs . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 93 98 104

8

Simply Connected Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The General Cauchy Closed Curve Theorem . . . . . . . . . . . . . . . . . . . . 8.2 The Analytic Function log z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 113 116

9

Isolated Singularities of an Analytic Function . . . . . . . . . . . . . . . . . . . . . 9.1 Classification of Isolated Singularities; Riemann’s Principle and the Casorati-Weierstrass Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Laurent Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

10 The Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Winding Numbers and the Cauchy Residue Theorem . . . . . . . . . . . . . 10.2 Applications of the Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129 129 135 141

6

7

11 Applications of the Residue Theorem to the Evaluation of Integrals and Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Evaluation of Definite Integrals by Contour Integral Techniques . . . 11.2 Application of Contour Integral Methods to Evaluation and Estimation of Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 65 68 74

82 90

117 120 126

143 143 143 151 158

Contents

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12 Further Contour Integral Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Shifting the Contour of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 An Entire Function Bounded in Every Direction . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161 161 164 167

13 Introduction to Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Conformal Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Special Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Schwarz-Christoffel Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 169 175 187 192

14 The Riemann Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Conformal Mapping and Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . 14.2 The Riemann Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Mapping Properties of Analytic Functions on Closed Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 195 200 204 213

15 Maximum-Modulus Theorems for Unbounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 A General Maximum-Modulus Theorem . . . . . . . . . . . . . . . . . . . . . . . 15.2 The Phragmén-Lindelöf Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215 215 218 223

16 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Poisson Formulae and the Dirichlet Problem . . . . . . . . . . . . . . . . . . . . 16.2 Liouville Theorems for Re f ; Zeroes of Entire Functions of Finite Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225 225

17 Different Forms of Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Analytic Functions Defined by Definite Integrals . . . . . . . . . . . . . . . . 17.3 Analytic Functions Defined by Dirichlet Series . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241 241 241 249 251 255

18 Analytic Continuation; The Gamma and Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Analytic Continuation of Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . 18.3 The Gamma and Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

257 257 257 263 265 271

233 238

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19 Applications to Other Areas of Mathematics . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1 A Variation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 The Fourier Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 An Infinite System of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 Applications to Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5 An Analytic Proof of The Prime Number Theorem . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

273 273 273 275 277 278 285 290

Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

Chapter 1

The Complex Numbers

Introduction √ Numbers of the form a + b −1, where a and b are real numbers—what we call complex numbers—appeared as early as the 16th century. Cardan (1501–1576) worked with complex numbers in solving quadratic and cubic equations. In the 18th century, functions involving complex numbers were found by Euler to yield solutions to differential equations. As more manipulations involving complex numbers were tried, it became apparent that many problems in the theory of real-valued functions could be most easily solved using complex numbers and functions. For all their utility, however, complex numbers enjoyed a poor reputation and were not generally considered legitimate numbers until the middle of the 19th century. Descartes, for example, rejected complex roots of equations and coined the term “imaginary” for such roots. Euler, too, felt that complex numbers “exist only in the imagination” and considered complex roots of an equation useful only in showing that the equation actually has no solutions. The wider acceptance of complex numbers is due largely to the geometric representation of complex numbers which was most fully developed and articulated by Gauss. He realized it was erroneous to assume “that there was some dark mystery in these numbers.” In the geometric representation, he wrote, one finds the “intuitive meaning of complex numbers completely established and more is not needed to admit these quantities into the domain of arithmetic.” Gauss’ work did, indeed, go far in establishing the complex number system on a firm basis. The first complete and formal definition, however, was given by his contemporary, William Hamilton. We begin with this definition, and then consider the geometry of complex numbers.

1.1 The Field of Complex Numbers We will see that complex numbers can be written in the form a + bi , where a and b are real numbers and i is a square root of −1. This in itself is not a formal definition, 1

2

1 The Complex Numbers

however, since it presupposes a system in which a square root of −1 makes sense. The existence of such a system is precisely what we are trying to establish. Moreover, the operations of addition and multiplication that appear in the expression a + bi have not been defined. The formal definition below gives these definitions in terms of ordered pairs. 1.1 Definition The complex field C is the set of ordered pairs of real numbers (a, b) with addition and multiplication defined by (a, b) + (c, d) = (a + c, b + d) (a, b)(c, d) = (ac − bd, ad + bc). The associative and commutative laws for addition and multiplication as well as the distributive law follow easily from the same properties of the real numbers. The additive identity, or zero, is given by (0, 0), and hence the additive inverse of (a, b) is (−a, −b). The multiplicative identity is (1, 0). To find the multiplicative inverse of any nonzero (a, b) we set (a, b)(x, y) = (1, 0), which is equivalent to the system of equations: ax − by = 1 bx + ay = 0 and has the solution

a −b , y= 2 . a 2 + b2 a + b2 Thus the complex numbers form a field. Suppose now that we associate complex numbers of the form (a, 0) with the corresponding real numbers a. It follows that x=

(a1 , 0) + (a2 , 0) = (a1 + a2 , 0) corresponds to a1 + a2 and that (a1 , 0)(a2 , 0) = (a1 a2 , 0) corresponds to a1 a2 . Thus the correspondence between (a, 0) and a preserves all arithmetic operations and there can be no confusion in replacing (a, 0) by a. In that sense, we say that the set of complex numbers of the form (a, 0) is isomorphic with the set of real numbers, and we will no longer distinguish between them. In this manner we can now say that (0, 1) is a square root of −1 since (0, 1)(0, 1) = (−1, 0) = −1

1.1 The Field of Complex Numbers

3

and henceforth (0, 1) will be denoted i . Note also that a(b, c) = (a, 0)(b, c) = (ab, ac), so that we can rewrite any complex number in the following way: (a, b) = (a, 0) + (0, b) = a + bi. We will use the latter form throughout the text. Returning to the question of square roots, there are in fact two complex square roots of −1: i and −i . Moreover, there are two square roots of any nonzero complex number a + bi . To solve (x + i y)2 = a + bi we set x 2 − y2 = a 2x y = b which is equivalent to 4x 4 − 4ax 2 − b2 = 0 y = b/2x. Solving first for x 2 , we find the two solutions are given by √ a + a 2 + b2 x =± 2 √ b −a + a 2 + b2 =± · (sign b) y= 2x 2 where

1 if b ≥ 0 sign b = −1 if b < 0.

E XAMPLE i. The two square roots of 2i are 1 + i and −1 − i . ii. The square roots of −5 − 12i are 2 − 3i and −2 + 3i .

♦

It follows that any quadratic equation with complex coefficients admits a solution in the complex field. For by the usual manipulations, az 2 + bz + c = 0 a, b, c ∈ C, a = 0

4


is seen to be equivalent to z+

b 2a

2 =

b 2 − 4ac , 4a 2

and hence has the solutions z=

−b ±

√ b2 − 4ac . 2a

(1)

In Chapter 5, we will see that quadratic equations are not unique in this respect: every nonconstant polynomial with complex coefficients has a zero in the complex field. One property of real numbers that does not carry over to the complex plane is the notion of order. We leave it as an exercise for those readers familiar with the axioms of order to check that the number i cannot be designated as either positive or negative without producing a contradiction.

1.2 The Complex Plane Thinking of complex numbers as ordered pairs of real numbers (a, b) is closely linked with the geometric interpretation of the complex field, discovered by Wallis, and later developed by Argand and by Gauss. To each complex number a + bi we simply associate the point (a, b) in the Cartesian plane. Real numbers are thus associated with points on the x-axis, called the real axis while the purely imaginary numbers bi correspond to points on the y-axis, designated as the imaginary axis. Addition and multiplication can also be given a geometric interpretation. The sum of z 1 and z 2 corresponds to the vector sum: If the vector from 0 to z 2 is shifted parallel to the x and y axes so that its initial point is z 1 , the resulting terminal point is z 1 + z 2 . If 0, z 1 and z 2 are not collinear this is the so-called parallelogram law; see below. y

y z1 + z2

z1

z1 z2

0

0

x

x

z 1 + z2

z2

The geometric method for obtaining the product z 1 z 2 is somewhat more complicated. If we form a triangle with two sides given by the vectors (originating from 0 to) 1 and z 1 and then form a similar triangle with the same orientation and the

1.2 The Complex Plane

5

vector z 2 corresponding to the vector 1, the vector which then corresponds to z 1 will be z 1 z 2 . This can be verified geometrically but will be most transparent when we introduce polar coordinates later in this section. For the moment, we observe that multiplication by i is equivalent geometrically to a counterclockwise rotation of 90◦. iz z2 z 1 z2 i

z

z1 0

0

1

1

With z = x + i y, the following terms are commonly used: Re z, the real part of z, is x; Im z, the imaginary part of z, is y (note that Im z is a real number); z¯ , the conjugate of z, is x − i y. Geometrically, z¯ is the mirror image of z reflected across the real axis.

z

Re z 0

–z

|z|, the absolute value or modulus of z, is equal to x 2 + y 2 ; that is, it is the length of the vector z. Note also that |z 1 − z 2 | is the (Euclidean) distance between z 1 and z 2 . Hence we can think of |z 2 | as the distance between z 1 + z 2 and z 1 and thereby obtain a proof of the triangle inequality: |z 1 + z 2 | ≤ |z 1 | + |z 2 |.

6


An algebraic proof of the inequality is outlined in Exercise 8. z1 + z2 |z2|

|z1 + z2|

z1 |z1| 0

Arg z, the argument of z, defined for z = 0, is the angle which the vector (originating from 0) to z makes with the positive x-axis. Thus Arg z is defined (modulo 2π) as that number θ for which cos θ =

Re z ; |z|

sin θ =

Im z . |z|

z |z|

|Im z|

θ 0

|Re z|

E XAMPLES i. The set of points given by the equation Re z > 0 is represented geometrically by the right half-plane. ii. {z : z = z¯ } is the real line. iii. {z : − θ < Arg z < θ } is an angular sector (wedge) of angle 2θ . iv. {z : |Arg z − π/2| < π/2} = {z : Im z > 0}. v. {z : |z + 1| < 1} is the disc of radius 1 centered at −1. ♦

1.2 The Complex Plane

7

(i)

0

θ

(iii)

0

(v) –1 0

8


A nonzero complex number is completely determined by its modulus and argument. If z = x + i y with |z| = r and Arg z = θ , it follows that x = r cos θ , y = r sin θ and z = r (cos θ + i sin θ ). We abbreviate cos θ + i sin θ as cis θ . In this context, r and θ are called the polar coordinates of z and r cis θ is called the polar form of the complex number z. This form is especially handy for multiplication. Let z 1 = r1 cis θ1 , z 2 = r2 cis θ2 . Then z 1 z 2 = r1r2 cis θ1 cis θ2 = r1r2 cis(θ1 + θ2 ), since (cos θ1 + i sin θ1 )(cos θ2 + i sin θ2 ) = (cos θ1 cos θ2 − sin θ1 sin θ2 ) + i (sin θ1 cos θ2 + cos θ1 sin θ2 ) = cos(θ1 + θ2 ) + i sin(θ1 + θ2 ) = cis(θ1 + θ2 ). Thus, if z is the product of two complex numbers, |z| is the product of their moduli and Arg z is the sum of their arguments (modulo 2π). (This can be used to verify the geometric construction for z 1 z 2 given at the beginning of this section.) Similarly z 1 /z 2 can be obtained by dividing the moduli and subtracting the arguments: r1 z1 = cis(θ1 − θ2 ). z2 r2 It follows by induction that if z = r cis θ and n is any integer, z n = r n cis nθ.

(1)

Identity (1) is especially handy for solving “pure” equations of the form z n = z 0 . E XAMPLE To find the cube roots of 1, we write z 3 = 1 in the polar form r 3 cis 3θ = 1 cis 0, which is satisfied if and only if r = 1, 3θ = 0 (modulo 2π). Hence the three solutions are given by z 1 = cis 0, z 2 = cis

2π 4π , z 3 = cis , 3 3

1.3 The Solution of the Cubic Equation

9

or in rectangular (x, y) coordinates √ √ 1 1 3 3 , z3 = − − i . z 1 = 1, z 2 = − + i 2 2 2 2 The polar form of the three cube roots reveals that they are the vertices of an equilateral triangle inscribed in the unit circle. Similarly the n-th roots of 1 are located at the vertices of the regular n-gon inscribed in the unit circle with one vertex at z = 1. For example, the fourth roots of 1 are ±1 and ±i . ♦

i

–1 1

–i

1.3 The Solution of the Cubic Equation As we mentioned at the beginning of this chapter, complex numbers were applied to the solution of quadratic and cubic equations as far back as the 16th century. While neither of these applications was sufficient to gain a wide acceptance of complex numbers, there was a fundamental difference between the two. In the case of quadratic equations, it may have seemed interesting that solutions could always be found among the complex numbers, but this was generally viewed as nothing more than an oddity at best. After all, if a quadratic equation (with real coefficients) had no real solutions, it seemed just as reasonable to simply say that there were no solutions as to describe so-called solutions in terms of some imaginary number. Cubic equations presented a much more tantalizing situation. For one thing, every cubic equation with real coefficients has a real solution. The fact that such a real solution could be found through the use of complex numbers showed that the complex numbers were at least useful, even if somewhat illegitimate. In fact, the solution of the cubic equation was followed by a string of other applications which demonstrated the uncanny ability of complex numbers to play a role in the solution of problems involving real numbers and functions. Let’s see how complex numbers were first applied to cubic equations. There is obviously no loss in assuming that the general cubic equation: ax 3 + bx 2 + cx + d = 0

10


has leading coefficient a = 1. The equation can then be further reduced to the simpler form: x 3 + px + q = 0 (1) if we change x into x − b3 . The first recorded solution for cubic equations involved a method for finding the real solution of the above “reduced” or “depressed” cubic in the form: x 3 + px = q (2) To the modern reader, of course, equation (2) is, for all practical purposes, identical to equation (1). But in the early 16th century, mathematicians were not entirely comfortable with negative numbers either, and it was assumed that the coefficients p and q in equation (2) denoted positive real numbers. In fact, in that case, f (x) = x 3 + px is a monotonically increasing function, so that equation (2) has exactly one positive real solution. To find that solution, del Ferro (1465–1526) suggested setting x = u + v, so that (2) could be rewritten as: u 3 + v 3 + (3uv + p)(u + v) = q

(3)

The solution to (3) can be found, then, by solving the pair of equations: 3uv+ p = 0 and u 3 + v 3 = q. Using the first equation to express v in terms of u, and substituting into the second equation leads to: u6 − u3q −

p3 =0 27

which is a quadratic equation for u 3 and has the solutions q ± q 2 + 4 p 3/27 3 . u = 2 The identical formula can be obtained for v 3 , and since u 3 + v 3 = q, 3 q + 3 q − q 2 + 4 p 3/27 q 2 + 4 p 3 /27 + . x =u+v = 2 2

(4)

or, as del Ferro would have written it to avoid the cube root of a negative number, 2 3 3 3 q + 4 p /27 + q q 2 + 4 p 3 /27 − q − x =u+v = 2 2 3 √ 3 √ For example, if p = 6 and q = 20, we find x = 6 3 + 10 − 6 3 − 10 or (check this!) x = 2. Although (4) was originally intended to be applied with p, q > 0, it can obviously be applied equally well for any values of p and q. Changing q into −q would simply cause the same change in x. For example, the unique real solution

1.3 The Solution of the Cubic Equation

11

of the equation x 3 + 6x = −20 is x = −2. Changing p into a negative number, however, can introduce complex values. To be precise, if q 2 + 4 p 3 /27 < 0; i.e., if 4 p 3 < −27q 2, equation (4) gives the solution as the sum of the cube roots of two complex conjugates. For example, if we apply (4) to the equation x 3 − 6x = 4, we obtain √ √ 3 3 x = 2 + 2i + 2 − 2i (5) Since we saw (in the last section) that we can calculate the three cube roots of any complex number using its polar form, and since the cube roots of a conjugate of any complex number are the conjugates of its cube roots, we realize that (5) actually does give the three real roots of x 3 − 6x = 4. To Cardan, however, who published formula (4) in his Ars Magna(1545), the case: 4 p 3 < −27q 2 presented a dilemma. We leave it as an exercise to verify that equation (2) has three real roots if and only if 4 p3 < −27q 2. Ironically, then, precisely in the case when all three solutions are real, if formula (4) is applicable at all, it gives the solutions in terms of cube roots of complex numbers! Moreover, Cardan was willing to try a direct approach to finding the cube roots of a complex number (as we found the square roots of any complex number in section 1), but solving the equation (x + i y)3 = a + bi by equating real and imaginary parts led to an equation no less complicated than the original cubic. Cardan, therefore, labeled this situation the “irreducible” case of the depressed cubic equation. Fortunately, however, the idea of applying (4) even in the “irreducible” case, was never laid to rest. Bombelli’s Algebra (1574) included the equation x 3 = 15x + 4, which led to the mysterious solution √ √ 3 3 x = 2 + 11i + 2 − 11i (6) By a direct approach, combined with the assumption that the cube roots in (6) would involve integral real and imaginary parts, Bombelli was able to show that formula (6) did “contain” the solution x = 4 in the form of (2 + i ) + (2 − i ). He did not suggest that (6) might also contain the other two real roots nor did he generalize the method. In fact, over a hundred years later, the issue was still not resolved. Thus Leibniz (1646–1716) continued to question how “a quantity could be real when imaginary or impossible numbers were used to express it”. But he too could not let the matter go. Among unpublished papers found after his death, there were several identities similar to √ √ 3 3 36 + −2000 + 36 − −2000 = −6 which he found by applying (4) to: x 3 − 48x − 72 = 0. So complex numbers maintained their presence, albeit as second-class citizens, in the world of numbers until the early 19th century when the spread of their geometric interpretation began the process of their acceptance as first-class citizens.

12


1.4 Topological Aspects of the Complex Plane I. Sequences and Series The concept of absolute value can be used to define the notion of a limit of a sequence of complex numbers. 1.2 Definition The sequence z 1 , z 2 , z 3 , . . . converges to z if the sequence of real numbers |z n − z| converges to 0. That is, z n → z if |z n − z| → 0. Geometrically, z n → z if each disc about z contains all but finitely many of the members of the sequence {z n }. Since |Re z|, |Im z| ≤ |z| ≤ |Re z| + |Im z|, z n → z if and only if Re z n → Re z and Im z n → Im z. E XAMPLES n 1. z n → 0 if |z| < 1 since |z n − 0| = |z| → 0. n n −i 1 2. → 1 since − 1 = = √ → 0. 2 n+i n +i n +i n +1

♦

1.3 Definition {z n } is called a Cauchy sequence if for each > 0 there exists an integer N such that n, m > N implies |z n − z m | < . 1.4 Proposition {z n } converges if and only if {z n } is a Cauchy sequence. Proof If z n → z, then Re z n → Re z, Im z n → Im z and hence {Re z n } and {Im z n } are Cauchy sequences. But since |z n − z m | ≤ |Re(z n − z m )| + |Im (z n − z m )| = |Re z n − Re z m | + |Im z n − Im z m |, {z n } is also a Cauchy sequence. Conversely, if {z n } is a Cauchy sequence so are the real sequences {Re z n } and {Im z n }. Hence both {Re z n } and {Im z n } converge, and thus {z n } converges. An infinite series ∞ k=1 z k is said to converge if the sequence {sn } of partial sums, defined by sn = z 1 + z 2 + · · · z n , converges. If so, the limit of the sequence is called

1.4 Topological Aspects of the Complex Plane

13

the sum of the series. The basic properties of infinite series listed below will be familiar from the theory of real series. i. ii. iii.

The sum and the difference of two convergent series are convergent. A necessary condition for ∞ n → 0 as n → ∞. k=1 z k to converge is that z∞ ∞ A sufficient condition for z to converge is that |z k | converges. When k k=1 k=1 ∞ ∞ k=1 |z k | converges, we will say k=1 z k is absolutely convergent.

Property (iii), which will be important in later chapters, follows from Proposition 1.4. For if ∞ k=1 |z k | converges and tn = |z 1 | + |z 2 | + · · · + |z n | then {tn } is a Cauchy sequence. But then so is the sequence {sn } given by sn = z 1 + z 2 + · · · + z n , since |sm − sn | = |z n+1 + z n+2 + · · · + z m | ≤ |z n+1 | + |z n+2 | + · · · + |z m | = |tm − tn | by the triangle inequality. Hence ∞ k=1 z k converges. E XAMPLES ∞ ik converges since 1. k=1 2 k +i k ∞

i 1 = √ 1 √ and since converges. k2 + i 4 4 k +1 k +1 k=1 2.

∞

k=1

1 diverges, since k +i

∞

k −i 1 1 = 2 , which implies that diverges. Re k +i k +1 k +i

♦

k=1

II. Classification of Sets in the Complex Plane We give some definitions relating to planar sets. 1.5 Definitions D(z 0 ; r ) denotes the open disc of radius r > 0 centered at z 0 ; i.e., D(z 0 ; r ) = {z : |z − z 0 | < r }. D(z 0 ; r ) is also called a neighborhood (or r -neighborhood) of z 0 . C(z 0 ; r ) is the circle of radius r > 0 centered at z 0 . A set S is said to be open if for any z ∈ S, there exists δ > 0 such that D(z; δ) ⊂ S. For any set S, S˜ = C\S denotes the complement of S; i.e., S˜ = {z ∈ C : z ∈ / S}. A set is closed set if its complement is open. Equivalently, S is closed if {z n } ⊂ S and z n → z imply z ∈ S. ∂ S, the boundary of S, is defined as the set of points whose δ-neighborhoods have ˜ for every δ > 0. a nonempty intersection with both S and S, ¯S, the closure of S, is given by S¯ = S ∪ ∂ S. S is bounded if it is contained in D(0; M) for some M > 0.

14


Sets that are closed and bounded are called compact. S is said to be disconnected if there exist two disjoint open sets A and B whose union contains S while neither A nor B alone contains S. If S is not disconnected, it is called connected. [z 1 , z 2 ] denotes the line segment with endpoints z 1 and z 2 . A polygonal line is a finite union of line segments of the form [z 0 , z 1 ] ∪ [z 1 , z 2 ] ∪ [z 2 , z 3 ] . . . ∪ [z n−1 , z n ]. If any two points of S can be connected by a polygonal line contained in S, S is said to be polygonally connected. z2

z1

It can be shown that a polygonally connected set is connected. The converse, however, is false. For example, the set of points z = x + i y with y = x 2 is clearly connected but is not polygonally connected since the set contains no straight line segments. In fact there are even connected sets whose points cannot be connected to one another by any curve in the set (see Exercise 23). On the other hand, for open sets, connectedness and polygonal connectedness are equivalent. 1.6 Definition An open connected set will be called a region. 1.7 Proposition A region S is polygonally connected. Proof Suppose z 0 ∈ S. Let A be the set of points of S which can be polygonally connected to z 0 in S and let B represent the set of points in S which cannot. Since any point z can be connected to any other point in D(z; δ), it follows that A is open. Similarly B is open. For if any point in a disc about z could be connected to z 0 , then z could be connected to z 0 . Now S is connected, S = A ∪ B and A is nonempty; hence we must conclude that B is empty. Finally, since every point in S can be connected to z 0 , every pair of points can be connected to each other by a polygonal line in S. III

Continuous Functions

1.8 Definition A complex valued function f (z) defined in a neighborhood of z 0 is continuous at z 0 if z n → z 0 implies that f (z n ) → f (z 0 ). Alternatively, f is continuous at z 0 if for

1.4 Topological Aspects of the Complex Plane

15

each > 0 there is some δ > 0 such that |z −z 0 | < δ implies that | f (z)− f (z 0 )| < . f is continuous in a domain D if for each sequence {z n } ⊂ D and z ∈ D such that z n → z, we have f (z n ) → f (z). If we split f into its real and imaginary parts f (z) = f (x, y) = u(x, y) + i v(x, y), where u and v are real-valued, it is clear that f is continuous if and only if u and v are continuous functions of (x, y). Thus, for example, any polynomial P(x, y) =

n m

akj x k y j

j =1 k=1

is continuous in the whole plane. Similarly x y 1 = 2 −i 2 z x + y2 x + y2 is continuous in the “punctured plane” {z : z = 0}. It follows also that the sum, product, and quotient (with nonzero denominator) of continuous functions are continuous. We say f ∈ C n if the real and imaginary parts of f both have continuous partial derivatives of the n-th order. A sequence of functions { f n } converges to f uniformly in D if for each > 0, there is an N > 0 such that n > N implies | f n (z) − f (z)| < for all z ∈ D. Again, by referring to the real and imaginary parts of { f n }, it is clear that the uniform limit of continuous functions is continuous. 1.9 M-Test Suppose f k is continuous in D, k = 1, 2, . . .. If | f k (z)| ≤ Mk throughout D and if ∞ ∞ M converges, then k k=1 k=1 f k (z) converges to a function f which is continuous in D. Proof

The convergence of ∞ k=1 f k (z) is immediate. Moreover, for each > 0, we can choose N so that ∞ n

≤ Mn+1 + Mn+2 + · · · < f (z) − = f (z) f (z) k k k=1

n+1

for n ≥ N. Hence the convergence is uniform and f is continuous.

16


E XAMPLE k f (z) = ∞ k=1 kz is continuous in D : |z| ≤ |kz k | ≤

1 2

since

∞

k k in D and k 2 2k k=1

converges. (See Exercise 21.)

♦

Recall that a continuous function maps compact/connected sets into compact/ connected sets. None of the other properties listed above, though, are preserved under continuous mappings. For example f (z) = Re z maps the open set C into the real line which is not open. The function g(z) = 1/z maps the bounded set: 0 < |z| < 1 onto the unbounded set: |z| > 1. Most of the key results in subsequent chapters will concern properties of (a certain class of) functions defined on a region. We note that, arguing as in the proof of Proposition 1.7, we could show that any two points in a region can be connected by a polygonal line containing only horizontal and vertical line segments. For future reference we will introduce the term polygonal path to denote such a polygonal line. One important result regarding real-valued functions on a region is given below. 1.10 Theorem Suppose u(x, y) has partial derivatives u x and u y that vanish at every point of a region D. Then u is constant in D. Proof Let (x 1 , y1 ) and (x 2 , y2 ) be two points of D. Then, as noted above, they can be connected by a polygonal path that is contained in D. Any two successive vertices of the path represent the end-points of a horizontal or vertical segment. Hence, by the Mean-Value Theorem for one real variable, the change in u between these vertices is given by the value of a partial derivative of u somewhere between the end-points times the difference in the non-identical coordinates of the endpoints. Since, however, u x and u y vanish identically in D, the change in u is 0 between each pair of successive vertices; hence u(x 1 , y1 ) = u(x 2 , y2 ).

1.5 Stereographic Projection; The Point at Infinity The complex numbers can also be represented by the points on the surface of a punctured sphere. Let 2

1 1 ; (1) = = (ξ, η, ζ ) : ξ 2 + η2 + ζ − 2 4

1.5 Stereographic Projection; The Point at Infinity

17

that is, let be the sphere in Euclidean (ξ, η, ζ ) space with distance 12 from (0, 0, 12 ). Suppose, moreover, that the plane ζ = 0 coincides with the complex plane C, and that the ξ and η axes are the x and y axes, respectively. To each (ξ, η, ζ ) ∈ we associate the complex number z where the ray from (0, 0, 1) through (ξ, η, ζ ) intersects C. This establishes a 1-1 correspondence, known as stereographic projection, between C and the points of other than (0, 0, 1). Formulas governing this correspondence can be derived as follows. Since (0, 0, 1), (ξ, η, ζ ) and (x, y, 0) are collinear, x y 1 = = ξ η 1−ζ ζ (0, 0, 1)

(ξ, η, ζ )

η=y

(0, 0, 0)

(x, y, 0)

ξ =x

so that x=

ξ ; 1−ζ

y=

η . 1−ζ

(2)

We leave it as an exercise to show that the equations (1) and (2) can be solved for ξ, η, ζ in terms of x, y as x 2 + y2 . (3) x 2 + y2 + 1 Now suppose that {σk } = {(ξk , ηk , ζk )} is a sequence of points of which converges to (0, 0, 1) and let {z k } be the corresponding sequence in C. By (2), ξ=

x ; x 2 + y2 + 1

η=

x 2 + y2 =

y ; x 2 + y2 + 1

ζ =

ξ 2 + η2 ζ , = 1−ζ (1 − ζ )2

so that as σk → (0, 0, 1), |z k | → ∞. Conversely, it follows from (3) that if |z k| → ∞, σk → (0, 0, 1). Loosely speaking, this suggests that the point (0, 0, 1) on corresponds to ∞ in the complex plane. We can make this more precise by formally adjoining to C a “point at infinity” and defining its neighborhoods as the sets in C corresponding to the spherical neighborhoods of (0, 0, 1). (See Exercise 24.)

18


While we will not examine the resulting “extended plane” in greater detail, we will adopt the following convention. 1.11 Definition We say {z k } → ∞ if |z k | → ∞; i.e., |z k | → ∞ if for any M > 0, there exists an integer N such that k > N implies |z k | > M. Similarly, we say f (z) → ∞ if | f (z)| → ∞. For future reference, and circles we note the connection between circles on in C. By a circle on , we mean the intersection of with a plane of the form Aξ + Bη +Cζ = D. According to (3), if S is such a circle and T is the corresponding set in C, (C − D)(x 2 + y 2 ) + Ax + By = D (4) for (x, y) ∈ T . Note that if C = D, (4) is the equation of a circle. If C = D, (4) represents a line. Since C = D if and only if S intersects (0, 0, 1), we have the following proposition. 1.12 Proposition Let S be a circle on

and let T be its projection on C. Then

a. if S contains (0, 0, 1), T is a line; b. if S doesn’t contain (0, 0, 1), T is a circle. The converse of Proposition 1.12 is also valid. We leave its proof as an exercise. (See Exercise 25.)

Exercises 1. Express in the form a + bi: 1 a. 6 + 2i √ 4 1 3 c. − + i 2 2

b.

(2 + i)(3 + 2i) 1−i

d. i 2 , i 3 , i 4 , i 5 , . . .

2. Find (in rectangular form) the two values of √ 3. Solve the equation z 2 + 32 iz − 6 i = 0.

√ −8 + 6 i.

4. Prove the following identities: a. z 1 + z 2 = z l + z 2 . b. z 1 z 2 = z 1 · z 2 . c. P(z) = P(¯z ), for any polynomial P with real coefficients. d. z¯¯ = z. 5. Suppose P is a polynomial with real coefficients. Show that P(z) = 0 if and only if P(¯z ) = 0 [i.e., zeroes of “real” polynomials come in conjugate pairs]. 6. Verify that |z 2 | = |z|2 using rectangular coordinates and then using polar coordinates.

Exercises

19

7. Show a. |z n | = |z|n . b. |z|2 = z z¯ . c. |Re z|, |Im z| ≤ |z| ≤ |Re z| + |Im z|. (When is equality possible?) 8. a. Fill in the details of the following proof of the triangle inequality: |z 1 + z 2 |2 = (z 1 + z 2 )(z 1 + z 2 ) = |z 1 |2 + |z 2 |2 + z 1 z 2 + z 1 z 2 = |z 1 |2 + |z 2 |2 + 2 Re(z 1 z 2 ) ≤ |z 1 |2 + |z 2 |2 + 2|z 1 ||z 2 | = (|z 1 | + |z 2 |)2 . b. When can equality occur? c. Show: |z 1 − z 2 | ≥ |z 1 | − |z 2 |. 9.* It is an interesting fact that a product of two sums of squares is itself a sum of squares. For example, (12 + 22 )(32 + 42 ) = 125 = 52 + 102 = 22 + 112 . a. Prove the result using complex algebra. That is, show that for any two pairs of integers {a, b} and {c, d}, we can find integers u, v with (a 2 + b2 )(c2 + d 2 ) = u 2 + v 2 b. Show that, if a, b, c, d are all nonzero and at least one of the sets {a 2 , b2 } and {c2 , d 2 } consists of distinct positive integers, then we can find u 2 , v 2 as above with u 2 and v 2 both positive. c. Show that, if a, b, c, d are all nonzero and both of the sets {a 2 , b2 } and {c2 , d 2 } consist of distinct positive integers, then there are two different sets {u 2 , v 2 } and {s 2 , t 2 } with (a 2 + b2 )(c2 + d 2 ) = u 2 + v 2 = s 2 + t 2 . d. Give a geometric interpretation and proof of the results in b) and c), above. 10.* Prove: |z 1 + z 2 |2 + |z 1 − z 2 |2 = 2(|z 1 |2 + |z 2 |2 ) and interpret the result geometrically. 11. Let z = x + iy. Explain the connection between Arg z and tan−1 (y/x). (Warning: they are not identical.) 12. Solve the following equations in polar form and locate the roots in the complex plane: a. z 6 = 1. b. z 4 = −1. √ c. z 4 = −1 + 3i. 13. Show that the n-th roots of 1 (aside from 1) satisfy the “cyclotomic” equation z n−1 + z n−2 + · · · + z + 1 = 0. [Hint: Use the identity z n − 1 = (z − 1)(z n−1 + z n−2 + · · · + 1).] 14. Suppose we consider the n − 1 diagonals of a regular n-gon inscribed in a unit circle obtained by connecting one vertex with all the others. Show that the product of their lengths is n. [Hint: Let the vertices all be connected to 1 and apply the previous exercise.]

20


15. Describe the sets whose points satisfy the following relations. of the sets are regions? Which z − 1 = 1. a. |z − i| ≤ 1. b. z + 1 c. |z − 2| > |z − 3|. e.

d. |z| < 1 and Im z > 0.

1 = z¯ . z

f. |z|2 = Im z.

g. |z 2 − 1| < 1. [Hint: Use polar coordinates.] 16.* Identify the set of points which satisfy a. |z| = Rez + 1 b. |z − 1| + |z + 1| = 4

c. z n−1 = z

17. Let Arg w denote that value of the argument between −π and π (inclusive). Show that Arg

z−1 z+1

=

π/2

if Im z > 0

−π/2 if Im z < 0

where z is a point on the unit circle |z| = 1. √ 18.* √ Find the three roots of x 3 − 6x = 4 by finding the three real-valued possibilities for 3 2 + 2i + 3 2 − 2i.(Note: You can find the three cube roots of 2 + 2i, or you can simplify the problem by first applying the identity: a + b = (a 3 + b3 )/(a 2 − ab + b2 ). 19.* Prove that x 3 + px = q has three real roots if and only if 4 p3 < −27q 2 . (Hint: Find the local minimum and local maximum values of x 3 + px − q.) 20.* a. Let P(z) = 1 + 2z + 3z 2 + · · · + nz n−1 . By considering (1 − z)P(z), show that all the zeroes of P(z) are inside the unit disc. b. Show that the same conclusion applies to any polynomial of the form: a0 +a1 z+a2 z 2 +···+an z n , with all ai real and 0 ≤ a0 ≤ a1 ≤ · · · ≤ an 21. Show that a. f (z) =

∞

b. g(z) =

∞

k k=0 kz is continuous in |z| < 1.

2 k=1 1/(k + z) is continuous in the right half-plane Re z > 0.

22. Prove that a polygonally connected set is connected. 23. Let

1 . S = x + iy : x = 0 or x > 0, y = sin x

Show that S is connected, even though there are points in S that cannot be connected by any curve in S. 24. Let S = {(ξ, η, ζ ) ∈ : ζ ≥ ζ0 }, where 0 < ζ0 < 1 and let T be the corresponding set in C. Show that T is the exterior of a circle centered at 0. 25. Suppose T ⊂ C. Show that the corresponding set S ⊂ is a. a circle if T is a circle. b. a circle minus (0, 0, 1) if T is a line. 26. Let P be a nonconstant polynomial in z. Show that P(z) → ∞ as z → ∞. 27. Suppose that z is the stereographic projection of (ξ, η, ζ ) and 1/z is the projection of (ξ , η , ζ ). a. Show that (ξ , η , ζ ) = (ξ, −η, 1 − ζ ). b. Show that the function 1/z, z ∈ C, is represented on by a 180◦ rotation about the diameter with endpoints (− 21 , 0, 12 ) and ( 12 , 0, 12 ). 28. Use exercise (27) to show that f (z) = 1/z maps circles and lines in C onto other circles and lines.

Chapter 2

Functions of the Complex Variable z

Introduction We wish to examine the notion of a “function of z” where z is a complex variable. To be sure, a complex variable can be viewed as nothing but a pair of real variables so that in one sense a function of z is nothing but a function of two real variables. This was the point of view we took in the last section in discussing continuous functions. But somehow this point of view is too general. There are some functions which are “direct” functions of z = x + i y and not simply functions of the separate pieces x and y. Consider, for example, the function x 2 − y 2 + 2i x y. This is a direct function of x + i y since x 2 − y 2 + 2i x y = (x + i y)2 ; it is the function squaring. On the other hand, the only slightly different-looking function x 2 + y 2 − 2i x y is not expressible as a polynomial in x + i y. Thus we are led to distinguish a special class of functions, those given by direct or explicit or analytic expressions in x + i y. When we finally do evolve a rigorous definition, these functions will be called the analytic functions. For now we restrict our attention to polynomials.

2.1 Analytic Polynomials 2.1 Definition A polynomial P(x, y) will be called an analytic polynomial if there exist (complex) constants αk such that P(x, y) = α0 + α1 (x + i y) + α2 (x + i y)2 + · · · + α N (x + i y) N . We will then say that P is a polynomial in z and write it as P(z) = α0 + α1 z + α2 z 2 + · · · + α N z N .

21

22

2 Functions of the Complex Variable z

x2

Indeed, x 2 − y 2 + 2i x y is analytic. On the other hand, as we mentioned above, + y 2 − 2i x y is not analytic, and we now prove this assertion. So suppose x + y − 2i x y ≡ 2

2

N

αk (x + i y)k .

k=0

Setting y = 0, we obtain x2 ≡

N

αk x k

k=0

or α0 + α1 x + (α2 − 1)x 2 + · · · + α N x N ≡ 0. Setting x = 0 gives α0 = 0; dividing out by x and again setting x = 0 shows α1 = 0, etc. We conclude that α1 = α3 = α4 = · · · = α N = 0 α2 = 1, and so our assumption that x 2 + y 2 − 2i x y ≡

N

αk (x + i y)k

k=0

has led us to x 2 + y 2 − 2i x y ≡ (x + i y)2 = x 2 − y 2 + 2i x y, which is simply false! A bit of experimentation, using the method described above (setting y = 0 and “comparing coefficients”) will show how rare the analytic polynomials are. A randomly chosen polynomial, P(x, y), will hardly ever be analytic. E XAMPLE x 2 + i v(x, y) is not analytic for any choice of the real polynomial v(x, y). For a polynomial in z can have a real part of degree 2 in x only if it is of the form az 2 + bz + c with a = 0. In that case, however, the real part must contain a y 2 term as well. ♦ Another Way of Recognizing Analytic Polynomials We have seen, in our method of comparing coefficients, a perfectly adequate way of determining whether a given polynomial is or is not analytic. This method, we point out, can be condensed to the statement: P(x, y) is analytic if and only if P(x, y) = P(x +i y, 0). Looking ahead to

2.1 Analytic Polynomials

23

the time we will try to extend the notion of “analytic” beyond the class of polynomials, however, we see that we can expect trouble! What is so simple for polynomials is totally intractable for more general functions. We can evaluate P(x +i y, 0) by simple arithmetic operations, but what does it mean to speak of f (x + i y, 0)? For example, if f (x, y) = cos x + i sin y, we observe that f (x, 0) = cos x. But what shall we mean by cos(x + i y)? What is needed is another means of recognizing the analytic polynomials, and for this we retreat to a familiar, real-variable situation. Suppose that we ask of apolynomial P(x, y) whether it is a function of the single variable x + 2y. Again the answer can be given in the spirit of our previous one, namely: P(x, y) is a function of x + 2y if and only if P(x, y) = P(x + 2y, 0). But it can also be given in terms of partial derivatives! A function of x + 2y undergoes the same change when x changes by ∈ as when y changes /2 and this means exactly that its partial derivative with the respect to y is twice its partial derivative with respect to x. That is, P(x, y) is a function of x + 2y if and only if Py = 2Px . Of course, the “2” can be replaced by any real number, and we obtain the more general statement: P(x, y) is a function of x + λy if and only if Py = λPx . Indeed for polynomials, we can even ignore the limitation that λ be real, which yields the following proposition. 2.2 Definition Let f (x, y) = u(x, y)+i v(x, y) where u and v are real-valued functions. The partial derivatives f x and f y are defined by u x + i v x and u y + i v y respectively, provided the latter exist. 2.3 Proposition A polynomial P(x, y) is analytic if and only if Py = i Px . Proof The necessity of the condition can be proven in a straightforward manner. We leave the details as an exercise. To show that it is also sufficient, note that if Py = i Px , the condition must be met separately by the terms of any fixed degree. Suppose then that P has n-th degree terms of the form Q(x, y) = C0 x n + C1 x n−1 y + C2 x n−2 y 2 + · · · + Cn y n . Since Q y = i Qx , C1 x n−1 + 2C2 x n−2 y + · · · + nCn y n−1 = i [nC0 x n−1 + (n − 1)C1 x n−2 y + · · · + Cn−1 y n−1 ].

24


Comparing coefficients, C1 = i nC0 = i C2 =

n C0 1

i (n − 1) n(n − 1) C1 = i 2 C0 = i 2 2 2

n C0 , 2

and in general Ck = i k

n C0 k

so that Q(x, y) =

n

Ck x n−k y k = C0

k=0

n

n

k

k=0

x n−k (i y)k = C0 (x + i y)n .

Thus P is analytic.

The condition f y = i f x is sometimes given in terms of the real and imaginary parts of f . That is, if f = u + i v, then fx = u x + i vx fy = uy + ivy and the equation f y = i f x is equivalent to the twin equations ux = vy;

u y = −v x .

(1)

These are usually called the Cauchy-Riemann equations. E XAMPLES 1. A non-constant analytic polynomial cannot be real-valued, for then both Px and Py would be real and the Cauchy-Riemann equations would not be satisfied. 2. Using the Cauchy-Riemann equations, one can verify that x 2 − y 2 + 2i x y is analytic while x 2 + y 2 − 2i x y is not. ♦ Finally, we note that polynomials in z have another property which distinguishes them as functions of z: they can be differentiated directly with respect to z. We will make this more precise below. 2.4 Definition A complex-valued function f , defined in a neighborhood of z, is said to be differentiable at z if lim

h→0

f (z + h) − f (z) h

exists. In that case, the limit is denoted f (z).

2.2 Power Series

25

It is important to note that in Definition 2.4, h is not necessarily real. Hence the limit must exist irrespective of the manner in which h approaches 0 in the complex plane. For example, f (z) = z¯ is not differentiable at any point z since f (z + h) − f (z) h¯ = h h which equals +1 if h is real and −1 if h is purely imaginary. 2.5 Proposition If f and g are both differentiable at z, then so are h1 = f + g h2 = f g and, if g(z) = 0, h3 =

f . g

In the respective cases, h 1 (z) = f (z) + g (z)

h 2 (z) = f (z)g(z) + f (z)g (z) h 3 (z) = [ f (z)g(z) − f (z)g (z)]/g 2 (z).

Proof Exercise 6.

2.6 Proposition If P(z) = α0 + α1 z + · · · + α N z N , then P is differentiable at all points z and P (z) = α1 + 2α2 z + · · · + Nα N z N−1 . Proof See Exercise 7.

2.2 Power Series We now consider a wider class of direct functions of z–those given by infinite polynomials or “power series” in z.

26


2.7 Definition

k A power series in z is an infinite series of the form ∞ k=0 Ck z . To study the convergence of a power series, we recall the notion of the lim of a positive real-valued sequence. That is,

lim an = lim

n→∞

n→∞

sup ak . k≥n

Since supk≥n ak is a non-increasing function of n, the limit always exists or equals +∞. The properties of the lim which will be of interest to us are the following. If limn→∞ an = L, i. for each N and for each > 0, there exists some k > N such that ak ≥ L − ; ii. for each > 0, there is some N such that ak ≤ L + for all k > N. iii. lim can = cL for any nonnegative constant c. 2.8 Theorem Suppose lim|Ck |1/k = L. 1. If L = 0, Ck z k converges for all z. 2. If L = ∞, Ck z k converges for z = 0 only. 3. If 0 < L < ∞, set R = 1/L. Then Ck z k converges for |z| < R and diverges for |z| > R. (R is called the radius of convergence of the power series.) Proof 1. L = 0. Since lim|Ck |1/k = 0, lim|Ck |1/k |z| = 0 for all z. Thus, for each z, there is some N such that k > N implies |Ck z k | ≤

1 , 2k

so that |Ck z k | converges; therefore, by the Absolute Convergence Test, Ck z k converges. 2. L = ∞. For any z = 0, |Ck |1/k ≥

1 |z|

for infinitely many values of k. Hence |Ck z k | ≥ 1, the terms of the series do not approach zero, and the series diverges. (The fact that the series converges for z = 0 is obvious.)

2.2 Power Series

27

3. 0 < L < ∞, R = 1/L. 1/ k |z| = Assume first that |z| < R and set |z| = R(1 − 2δ). Then since k| lim|C 1/k k (1 − 2δ), |Ck | |z| < 1 − δ for sufficiently large k and Ck z is absolutely convergent. On the order hand, if |z| > R, lim|Ck |1/k |z| > 1, so that for infinitely many values of k, Ck z k has absolute value greater than 1 and Ck z k diverges. k Note that if ∞ k=0 Ck z has radius of convergence R, the series converges uniformly in any smaller disc: |z| ≤ R − δ. For then ∞

k=0

|Ck z k | ≤

∞

|Ck |(R − δ)k ,

k=0

which also converges. Hence a power series is continuous throughout its domain of convergence. (See Theorem 1.9.) All three cases above can be combined by noting that a power series always converges inside a disc of radius R = 1/lim|Ck |1/k . Here R = 0 means that the series converges at z = 0 only and R = ∞ means that the series converges for all z. In the cases where 0 < R < ∞, while the theorem assures us that the series diverges for |z| > R, it says nothing about the behavior of the power series on the circle of convergence |z| = R. As the following examples demonstrate, the series may converge for all or some or none of the points on the circle of convergence. E XAMPLES n 1. Since n 1/n → 1, ∞ n=1 nz converges for |z| < 1 and diverges for |z| > 1. The series diverges for |z| = 1 for then |nz n | = n → ∞. (See Exercise 8.) ∞ also n 2 2. n=1 (z /n ) also has radius of convergence equal to 1. In this case, however, the series converges for all points z on the unit circle since n z = 1 for |z| = 1. n2 n2 ∞ n 3. n=1 (z /n) has radius of convergence equal to 1. In this case, the series converges at points of the unit circle except z = 1. (See Exercise 12.) all ∞ n 4. n=0 (z /n!) converges for all z since 1 → 0. (n!)1/n 5.

(See ∞ Exercise 13.)n n n n=0 [1 + (−1) ] z has radius of convergence lim 2 = 2.

1 2

since lim[1 + (−1)n ] =

28


∞

2

z n = 1 + z + z 4 + z 9 + z 16 + · · · has radius of convergence 1. In this case lim|Cn |1/n = lim 1 = 1. 7. Any series of the form Cn z n with Cn = ±1 for all n has radius of convergence equal to 1. ♦ 6.

n=0

It is easily seen that the sum of two power series is convergent wherever both of the original two power series are convergent. In fact, it follows directly from the definition of infinite series that ∞ ∞ ∞

(an + bn )z n = an z n + bn z n . n=0

∞

n=0

n=0

∞

∞ n n n Similarly if n=0 ann z = A and n=0 bn z = B, the Cauchy product n=0 cn z defined by cn = k=0 ak bn−k converges for appropriate values of z to the product AB. The proof is the same as that for “real” power series and is outlined in Exercises 17 and 18.

2.3 Differentiability and Uniqueness of Power Series We now show that power like polynomials, are differentiable functions of series, n converges in some disc D(0; R), R > 0. Then the z. Suppose then that C z n series nCn z n−1 obtained by differentiating Cn z n term by term is convergent in D(0; R), since lim|nCn |1/(n−1) = lim(|nCn |1/n )n/(n−1) = lim|Cn |1/n . 2.9 Theorem

n Suppose f (z) = ∞ n=0 Cn z converges for |z| < R. Then f (z) exists and equals ∞ n−1 throughout |z| < R. n=0 nCn z Proof We will prove the theorem in two stages. First, we will assume that R = ∞, then we will consider the more general situation. Of course, the second case contains the first, so the eager reader may skip the first proof. We include it since it contains the key ideas with less cumbersome details. n Case (1): Assume ∞ n=0 Cn z converges for all z. Then ∞

f (z + h) − f (z) [(z + h)n − z n ] = Cn h h n=0

and ∞

∞

n=0

n=2

f (z + h) − f (z)

− nCn z n−1 = C n bn h

2.3 Differentiability and Uniqueness of Power Series

29

where (z + h)n − z n − nz n−1 h n n

n n h k−1 z n−k ≤ |h| |z|n−k = |h|(|z| + 1)n = k k

bn =

k=2

k=0

for |h| ≤ 1. Hence, for |h| ≤ 1, ∞ ∞ f (z + h) − f (z)

n−1 − nCn z ≤ |h| |Cn |(|z| + 1)n ≤ A|h| h n=0

since

∞

n=0 |Cn |z

n

n=0

converges for all z. Letting h → 0, we conclude that

f (z) = nCn z n−1 .

Case (2): 0 < R < ∞. Let |z| = R − 2δ, δ > 0, and assume |h| < δ. Then |z + h| < R and, as in the previous case. we can write ∞

∞

n=0

n=2

f (z + h) − f (z)

− nCn z n−1 = C n bn , h where bn =

n

n k=2

k

h k−1 z n−k .

If z = 0, bn = h n−1 and the proof follows easily. Otherwise, to obtain a useful estimate for bn we must be a little more careful. Note then that n(n − 1) · · · (n − k + 1) n n 2 = for k ≥ 2. ≤n k k−2 k! Hence, for z = 0, n n 2 |h|

n |bn | ≤ |h|k−2 |z|n−(k−2) k−2 |z|2 k=2 n n 2 |h| n |h| j |z|n− j ≤ j |z|2 j =0

=

n 2 |h|

(|z| + |h|)n |z|2 n 2 |h| ≤ (R − δ)n |z|2

30

and


∞ ∞ f (z + h) − f (z)

|h| 2 − nCn z n−1 ≤ 2 n |Cn |(R − δ)n ≤ A|h|, |z| h n=0

n=0

∞

n also converges for |z| < R. Again, since z = 0 is fixed and since n=0 n 2 |C n |z ∞ letting h → 0, we conclude that f (z) = n=0 nCn z n−1 .

E XAMPLE n f (z) = ∞ n=0 (z /n!) is convergent for all z and, according to Theorem 2.9, f (z) =

∞

nz n−1 n=0

n!

=

∞ n

z n=0

n!

= f (z). ♦

2.10 Corollary Power series are infinitely differentiable within their domain of convergence. Proof

n−1 which has the same radius Applying the above results to f (z) = ∞ n=0 nCn z of convergence as f, we see that f is twice differentiable. By induction, f (n) is differentiable for all n. 2.11 Corollary n If f (z) = ∞ n=0 Cn z has a nonzero radius of convergence, Cn =

f (n) (0) f or all n. n!

Proof By definition f (0) = C0 . Differentiating the power series term-by-term gives f (z) = C1 + 2C2 z + 3C3 z 2 + · · · so that f (0) = C1 . Similarly f (n) (z) = n!Cn + (n + 1)!Cn+1 z + and the result follows by setting z = 0.

(n + 2)! Cn+2 z 2 + · · · , 2!

2.3 Differentiability and Uniqueness of Power Series

31

According to Corollary 2.11, if a power series is equal to zero throughout a neighborhood of the origin, it must be identically zero. For then all its derivatives at the origin–and hence all the coefficients of the power series–would equal 0. By the same reasoning, if a power series were equal to zero throughout an interval containing the origin, it would be identically zero. An even stronger result is proven below. 2.12 Uniqueness Theorem for Power Series n Suppose ∞ n=0 Cn z is zero at all points of a nonzero sequence {z k } which converges to zero. Then the power series zero. is identically [Note: If we set f (z) = Cn z n , it follows from the continuity of power series that f (0) = 0. We can show by a similar argument that f (0) = 0; however, a slightly different argument is needed to show that the higher coefficients are also 0.] Proof Let f (z) = C0 + C1 z + C2 z 2 + · · · . By the continuity of f at the origin C0 = f (0) = lim f (z) = lim f (z k ) = 0. k→∞

z→0

But then

f (z) = C1 + C2 z + C3 z 2 + · · · z is also continuous at the origin and g(z) =

C1 = g(0) = lim

z→0

f (z) f (z k ) = lim = 0. k→∞ z k z

Similarly, if C j = 0 for 0 ≤ j < n, then Cn = lim

z→0

f (z) f (z k ) = lim = 0, n k→∞ z kn z

so that the power series is identically zero.

2.13 Corollary If a power series equals zero at all the points of a set with an accumulation point at the origin, the power series is identically zero. Proof Exercise 18. The Uniqueness Theorem derives its name from the following corollary.

32


2.14 Corollary If an z n and bn z n converge and agree on a set of points with an accumulation point at the origin, then an = bn for all n. Proof Apply 2.13 to the difference:

(an − bn )z n .

Power Series Expansion about z = α All of the previous results on power series are easily adapted to power series of the form

Cn (z − α)n . By the simple substitution w = z − α, we see, for example, that series of the above form converge in a disc of radius R about z = α and are differentiable throughout |z − α| < R where R = 1/lim|Cn |1/n . (See Exercises 22 and 23.)

Exercises 1. Complete the proof of Proposition 2.3 by showing that for an analytic polynomial P, Py = i Px . [Hint: Prove it first for the monomials.] 2.* a. Suppose f (z) is real-valued and differentiable for all real z. Show that f (z) is also real-valued for real z. b. Suppose f (z) is real-valued and differentiable for all imaginary points z. Show that f (z) is imaginary for at all imaginary points z. 3. By comparing coefficients or by use of the Cauchy-Riemann equations, determine which of the following polynomials are analytic. a. P(x + iy) = x 3 − 3xy 2 − x + i(3x 2 y − y 3 − y). b. P(x + iy) = x 2 + iy 2 . c. P(x + iy) = 2xy + i(y 2 − x 2 ). 4. Show that no nonconstant analytic polynomial can take imaginary values only. 5. Find the derivative P (z) of the analytic polynomials in (3). Show that in each case P (z) = Px . Explain. 6. Prove Proposition 2.5 by arguments analogous to those of real-variable calculus. 7. Prove Proposition 2.6. [Hint: Prove it for monomials and apply Proposition 2.5.] 8. Show Sn = n 1/n → 1 as n → ∞ by considering log Sn . 9. Find the radius of convergence of the following power series: ∞ ∞ n ! n n b. a. n=0 z , n=0 (n + 2 )z . n 10. Suppose cn z has radius of convergence R. Find the radius of convergence of p b. |cn |z n , a. n c zn , 2 nn c. cn z .

Exercises

33

11. Suppose an z n and bn z n have radii of convergence R1 and R2 , respectively. What can be said about the radius of convergence of (an + bn )z n ? Show, by example, that the radius of convergence of the latter may be greater than R1 and R2 . n 12. Show that ∞ n=1 (z /n) converges at all points on the unit circle except z = 1. [Hint: Let z = cis θ and analyze the real and imaginary parts of the series separately.] 13. a. Suppose {an } is a sequence of positive real numbers and lim

n→∞

an+1 = L. an

1/n

Show then that limn→∞ an = L. b. Use the result above to prove

1 1/n → 0. n!

14. Use Exercise (13a) to find the radius of convergence of ∞ (−1)n z n ∞ z 2n+1 a. , b. n=0 n=0 (2n + 1)! , n! c.

∞

n!z n n=1 n n ,

d.

15.* Find the radius of convergence of a. sin n z n ,

b.

∞

n=0

2n z n . n!

−n 2 n e z ,

2 16.* Find the radius of convergence of cn z n if c2k = 2k ; c2k−1 = (1 + 1/k)k , k = 1, 2, ... ∞ ∞ 17. Suppose k=0 ak = A and k=0 bk = B. Suppose further that each of the series is absolutely convergent. Show that if k

a j bk− j ck = j =0

then

Outline: Use the fact that

∞

|ak | and

ck = AB.

k=0

|bk | converge to show that

dk =

k

dk converges where

|a j ||bk− j |.

j =0

In particular, dn+1 + dn+2 + · · · → 0 as n → ∞. Note then that if An = a0 + a1 + · · · + an Bn = b0 + b1 + · · · + bn Cn = c0 + c1 + · · · + cn , An Bn = Cn + Rn , where |Rn | ≤ dn+1 + dn+2 + · · ·+ d2n , and the result follows by letting n → ∞. 18. Suppose an z n and bn z n have radii of convergence R1 and R2 respectively. Show that the Cauchy product cn z n converges for |z| < min(R1 , R2 ).

34


19. a. Using the identity (1 − z)(1 + z + z 2 + · · · + z N ) = 1 − z N +1 show that ∞

zn =

n=0

1 for |z| < 1. 1−z

n b. By taking the Cauchy product of n=0 z n with itself, find a closed form for ∞ n=0 nz . 20. Prove Corollary 2.13 by showing that if a set S has an accumulation point at 0, it contains a sequence of nonzero terms which converge to 0. n 21. Show that there is no power series f (z) = ∞ n=0 C n z such that ∞

i. f (z) = 1 for z = 12 , 13 , 14 , . . . , ii. f (0) > 0. 22. Assume lim|Cn |1/n < ∞. Show that if we set f (z) =

∞

Cn (z − α)n ,

n=0

then Cn = 23. Find the domain of convergence of ∞ n a. n=0 n(z − 1) , ∞ 2 n c. n=0 n (2z − 1) .

b.

f (n) (α) . n! ∞

n=0

(−1)n (z + 1)n , n!

Chapter 3

Analytic Functions

3.1 Analyticity and the Cauchy-Riemann Equations The direct functions of z which we have studied so far—polynomials and convergent power series—were shown to be differentiable functions of z. We now take a closer look at the property of differentiability and its relation to the Cauchy-Riemann equations. As we mentioned earlier (after Definition 2.4), if f is differentiable, lim

h→0

f (z + h) − f (z) h

must exist regardless of the manner in which h approaches 0 through complex values. An immediate consequence is that the partial derivatives of f must satisfy the CauchyRiemann equations. 3.1 Proposition If f = u + i v is differentiable at z, f x and f y exist there and satisfy the CauchyRiemann equation f y = i fx or, equivalently, ux = vy u y = −v x . Proof Suppose first that h → 0 through real values. Then f (x + h, y) − f (x, y) f (z + h) − f (z) = → fx . h h 35

36

3 Analytic Functions

On the other hand, if h → 0 along the imaginary axis, h = i η and fy f (x, y + η) − f (x, y) f (z + h) − f (z) = → . h iη i (See Exercise 1.) Since the two limits must be equal, fy = i fx . As we mentioned in Chapter 2, setting f = u + i v, the equation f y = i f x takes the form u y + i v y = i (u x + i v x ) and hence ux = vy u y = −v x .

The converse of the above proposition is not true. There are functions which are not differentiable at a point despite the fact that the partial derivatives exist and satisfy the Cauchy-Riemann equations there. For example, consider ⎧ ⎨ x y(x + i y) z = 0 x 2 + y2 f (z) = f (x, y) = ⎩ 0 z = 0. f = 0 on both axes so that f x = f y = 0 at the origin but lim

z→0

f (z) − f (0) xy = lim 2 (x,y)→(0,0) x + y 2 z

does not exist. For on the line y = αx α f (z) − f (0) ≡ for z = 0 z 1 + α2 and hence the limit depends on α! The following partial converse, however, is true. 3.2 Proposition Suppose f x and f y exist in a neighborhood of z. Then if f x and f y are continuous at z and f y = i f x there, f is differentiable at z. Proof Let f = u + i v, h = ξ + i η.

3.1 Analyticity and the Cauchy-Riemann Equations

37

We will show that f (z + h) − f (z) → f x (z) = u x (z) + i v x (z) h as h → 0. By the Mean-Value Theorem (for real functions of a real variable) u(x + ξ, y + η) − u(x, y) u(z + h) − u(z) = h ξ + iη u(x + ξ, y + η) − u(x + ξ, y) = ξ + iη u(x + ξ, y) − u(x, y) + ξ + iη η = u y (x + ξ, y + θ1 η) ξ + iη ξ u x (x + θ2 ξ, y), + ξ + iη and η v(z + h) − v(z) = v y (x + ξ, y + θ3 η) h ξ + iη ξ v x (x + θ4 ξ, y) + ξ + iη for some θk , 0 < θk < 1, k = 1, 2, 3, 4. Thus η f (z + h) − f (z) = [u y (z 1 ) + i v y (z 2 )] h ξ + iη ξ [u x (z 3 ) + i v x (z 4 )] + ξ + iη where |z k − z| → 0 as h → 0, k = 1, 2, 3, 4. Since f y = i f x at z we can subtract f x (z) in the form of ξ η fy + fx ξ + iη ξ + iη to obtain η f (z + h) − f (z) − f x (z) = [(u y (z 1 ) − u y (z)) + i (v y (z 2 ) − v y (z))] h ξ + iη ξ [(u x (z 3 ) − u x (z)) + i (v x (z 4 ) − v x (z))]. + ξ + iη

38

Finally, since


η ξ , ξ + i η ξ + i η ≤ 1,

while each of the bracketed expressions approaches 0 as h → 0, lim

h→0

f (z + h) − f (z) = f x (z). h

E XAMPLE Let f (z) = |z|2 = x 2 + y 2 . Then f x = 2x, f y = 2y so that f has continuous partial derivatives for all z. By the previous proposition, then f is differentiable if and only if f y = i f x . Hence f is differentiable only at the point z = 0. ♦ To avoid pathologies such as that given in the example above, we adopt the following definition. 3.3 Definition f is analytic at z if f is differentiable in a neighborhood of z. Similarly, f is analytic on a set S if f is differentiable at all points of some open set containing S. Note that this definition is consistent with Definition 2.1 for analytic polynomials. For we have already noted (Proposition 2.6) that “polynomials in z” are everywhere differentiable. Conversely, if a polynomial P is analytic at a point z, its partial derivatives must satisfy the Cauchy-Riemann equations throughout a neighborhood of z. Hence, as in Proposition 2.3, it follows that P must be a “polynomial in z.” Functions, such as polynomials or everywhere convergent power series, that are everywhere differentiable are called entire functions. As we saw in Propositions 2.5 and 2.6, many of the properties of differentiability are analogous to those of differentiable functions of a real variable. Similarly, the composition of differentiable functions is differentiable (see Exercise 3). As in the “real” case, the inverse of a differentiable function need not even be continuous. Under the appropriate hypothesis, however, we can establish the differentiability of inverse functions. 3.4 Definition Suppose that S and T are open sets and that f is 1-1 on S with f (S) = T . g is the inverse of f on T if f (g(z)) = z for z ∈ T . g is the inverse of f at z 0 if g is the inverse of f in some neighborhood of z 0 . Note that an inverse function must be 1-1 for if f −1 (z) = f −1 (z 0 ), f ( f −1 (z)) = f ( f −1 (z 0 )); i.e., z = z 0 .

3.1 Analyticity and the Cauchy-Riemann Equations

39

3.5 Proposition Suppose that g is the inverse of f at z 0 and that g is continuous there. If f is differentiable at g(z 0 ) and if f (g(z 0 )) = 0, then g is differentiable at z 0 and g (z 0 ) =

1 . f (g(z 0 ))

Proof g(z) − g(z 0 ) 1 = f (g(z)) − f (g(z 0 )) z − z0 g(z) − g(z 0 ) for all z = z 0 in a neighborhood of z 0 . Since g is continuous at z 0 , g(z) → g(z 0 ) as z → z 0 , and by the differentiability of f, lim

z→z 0

1 g(z) − g(z 0 ) . = z − z0 f (g(z 0 ))

As we shall see in the coming chapters, the property of analyticity is a very far-reaching one. Some immediate consequences are proven below. 3.6 Proposition If f = u + i v is analytic in a region D and u is constant, then f is constant. Proof Since u is constant, u x = u y = 0; therefore, by the Cauchy-Riemann equations, v x = v y = 0. According to Theorem 1.10, u and v are each constant in D; hence f is constant. 3.7 Proposition If f is analytic in a region and if | f | is constant there, then f is constant. Proof If | f | = 0, the proof is immediate. Otherwise u 2 + v 2 ≡ C = 0. Taking the partial derivatives with respect to x and y, we see that uu x + vv x ≡ 0 uu y + vv y ≡ 0.

40


Making use of the Cauchy-Riemann equations, we obtain uu x − vu y ≡ 0 vu x + uu y ≡ 0, so that (u 2 + v 2 )u x ≡ 0 and u x = v y ≡ 0. Similarly, u y and v x are identically zero, hence f is constant.

3.2 The Functions e z , sin z, cos z We wish to define an exponential function of the complex variable z; that is, we seek an analytic function f such that f (z 1 + z 2 ) = f (z 1 ) f (z 2 ), f (x) = e

x

for all real x.

(1) (2)

According to (1) and (2) we must have f (z) = f (x + i y) = f (x) f (i y) = e x f (i y). Setting f (i y) = A(y) + i B(y), it follows that f (z) = e x A(y) + i e x B(y). For f to be analytic, the Cauchy-Riemann equations must be satisfied; therefore A(y) = B (y) and A (y) = −B(y), so that A = −A. Thus we consider A(y) = α cos y + β sin y B(y) = −A (y) = −β cos y + α sin y. Since f (x) = e x , however, A(0) = α = 1 and B(0) = −β = 0, so that, finally, we are led to examine f (z) = e x cos y + i e x sin y. Indeed, it is easy to verify that f is an entire function with the desired properties (1) and (2). (See Exercise 11.) Hence f is an entire “extension” of the real exponential function and we write f (z) = e z . The following properties of e z are easily proven: i. |e z | = e x . ii. e z = 0. This follows from (i) since e x = 0. Also, according to (1), above, e z e−z =e0 =1. iii. eiy = cis y. iv. e z = α has infinitely many solutions for any α = 0.

Exercises

41

Proof Set α = r cis θ = r eiθ , r > 0. Since e z = e x eiy , we will have e z = α if x = log r and eiy = eiθ . Hence e z = α for all points z = x + i y with x = log r , y = Arg α = θ ± 2kπ, k = 0, 1, 2 . . .. v. (e z ) = e z . Recall that (e z ) = (e z )x = e z . To define sin z and cos z, note that for real y eiy = cos y + i sin y e−iy = cos y − i sin y so that sin y =

1 iy (e − e−iy ) 2i

and

1 iy (e + e−iy ). 2 Thus we can define entire extensions of sin x and cos x by setting cos y =

1 iz (e − e−iz ) 2i 1 cos z = (eiz + e−iz ). 2 sin z =

Many of the familiar properties of the sin and cos functions remain valid in the larger setting of the complex plane. For example, sin 2z = 2 sin z cos z sin2 z + cos 2 z = 1 (sin z) = cos z. These identities are easily verified and are left as an exercise. Moreover, in Section 6.3, we will see that, in general, functional equations of the above form, known to be true on the real axis, remain valid throughout the complex plane. On the other hand, unlike sin x, sin z is not bounded in modulus by 1. For example, | sin 10i | = 12 (e10 − e−10 ) > 10, 000.

Exercises 1. Show that f x = lim

h→0 h real

provided the limits exist.

f (z + h) − f (z) f (z + ih) − f (z) ; f y = lim , h h h→0 h real

42


2. a. Show that f (z) = x 2 + iy 2 is differentiable at all points on the line y = x. b. Show that it is nowhere analytic. 3. Prove that the composition of differentiable functions is differentiable. That is, if f is differentiable at z, and if g is differentiable at f (z), then g ◦ f is differentiable at z. [Hint: Begin by noting g( f (z + h)) − g( f (z)) = [g ( f (z)) + ][ f (z + h) − f (z)] where → 0 as h → 0.]

√ 4. Suppose that g is a continuous “ z” (i.e., g 2 (z) = z) in some neighborhood of z. Verify that √ g (z) = 1/2 z. [Hint: Use g 2 (z) − g 2 (z 0 ) 1= z − z0 to evaluate lim

z→z 0

g(z) − g(z 0 ) .] z − z0

5. Suppose f is analytic in a region and f ≡ 0 there. Show that f is constant. 6. Assume that f is analytic in a region and that at every point, either f = 0 or f = 0. Show that f is constant. [Hint: Consider f 2 .] 7. Show that a nonconstant analytic function cannot map a region into a straight line or into a circular arc. 8. Find all analytic functions f = u + iv with u(x, y) = x 2 − y 2 . 9. Show that there are no analytic functions f = u + iv with u(x, y) = x 2 + y 2 . 10. Suppose f is an entire function of the form f (x, y) = u(x) + iv(y). Show that f is a linear polynomial. 11. a. Show that ez is entire by verifying the Cauchy-Riemann equations for its real and imaginary parts. b. Prove: ez 1 +z 2 = ez 1 ez 2 . 12. Show: |ez | = e x . 13. Discuss the behavior of ez as z → ∞ along the various rays from the origin. 14. Find all solutions of a. ez = 1, c. ez = −3,

b. ez = i, d. ez = 1 + i.

15. Verify the identities a. sin 2z = 2 sin z cos z, b. sin2 z + cos2 z = 1, c. (sin z) = cos z. 16.* Show that a. sin( π2 + iy) = 12 (e y + e−y ) = cosh y b. | sin z| ≥ 1 at all points on the square with vertices ±(N + 12 )π ± (N + 12 )π i, for any positive integer N. c. | sin z| → ∞, as Imz = y → ±∞.

Exercises

43

17. Find (cos z) . 18. Find sin−1 (2)– that is, find the solutions of sin z = 2. [Hint: First set w = eiz and solve for ω.] 19.* Find all solutions of the equation:

z

ee = 1. 20. Show that sin(x + iy) = sin x cosh y + i cos x sinh y. 21. Show that the power series f (z) = 1 + z +

∞ n

z2 z + ... = 2! n! n=0

is equal to ez . [Hint: First show that f (z) f (w) = f (z + w), then show f (x) = e x f (iy) = cos y + i sin y using the power series representations for the real functions e x , cos x, sin x.] 22. Show: g(z) = z −

z5 z3 + − +... 3! 5!

is equal to sin z. [Hint: Use the power series representation for ez given in (21) to show that g(z) = 23. Find a power series representation for cos z.

1 iz (e − e−iz ).] 2i

Chapter 4

Line Integrals and Entire Functions

Introduction Recall that, according to Theorem 2.9, an everywhere convergent power series represents an entire function. Our main goal in the next two chapters is the somewhat surprising converse of that result: namely, that every entire function can be expanded as an everywhere convergent power series. As an immediate corollary, we will be able to prove that every entire function is infinitely differentiable. To arrive at these results, however, we must begin by discussing integrals rather than derivatives.

4.1 Properties of the Line Integral 4.1 Definition Let f (t) = u(t) + i v(t) be any continuous complex-valued function of the real variable t, a ≤ t ≤ b. b b b f (t)dt = u(t)dt + i v(t)dt. a

a

a

4.2 Definition a. Let z(t) = x(t) + i y(t), a ≤ t ≤ b. The curve determined by z(t) is called piecewise differentiable and we set z˙ (t) = x (t) + i y (t) if x and y are continuous on [a, b] and continuously differentiable on each subinterval [a, x 1 ], [x 1 , x 2 ], . . . , [x n−1 , b] of some partition of [a, b]. b. The curve is said to be smooth if, in addition, z˙ (t) = 0 (i.e., x (t) and y (t) do not both vanish) except at a finite number of points.

45

46

4 Line Integrals and Entire Functions

Throughout the remainder of the text, all curves will be assumed to be smooth unless otherwise stated. Finally, we define the important concept of a line integral. 4.3 Definition Let C be a smooth curve given by z(t), a ≤ t ≤ b, and suppose f is continuous at all the points z(t). Then, the integral of f along C is

b

f (z)dz =

f (z(t))˙z (t)dt.

a

C

Note that the integral along the curve C depends not only on the points of C but on the direction as well. However, we will show that it is independent of the particular parametrization. Intuitively, if z(t), a ≤ t ≤ b, and ω(t), c ≤ t ≤ d, trace the same curve in the same direction, then λ = z −1 ◦ ω will be a 1-1 mapping of [c, d] onto [a, b] such that ω(t) = z(λ(t)). (1) However, if z is not 1-1, it is difficult to define z −1 . Instead, we take the existence of some λ that satisfies (1) as the definition for equivalent curves. C

z a

ω b

c

d

4.4 Definition The two curves C1 : z(t), a ≤ t ≤ b and C2 : ω(t), c ≤ t ≤ d are smoothly equivalent if there exists a 1-1 C 1 mapping λ(t) : [c, d] → [a, b] such that λ(c) = a, λ(d) = b, λ (t) > 0 for all t, and ω(t) = z(λ(t)). (It is easy to verify that the above is an equivalence relation. See Exercise 1.) 4.5 Proposition If C1 and C2 are smoothly equivalent, then f = C1

f. C2

4.1 Properties of the Line Integral

47

Proof Suppose f (z) = u(z) + i v(z), C1 and C2 as above. Then, by definition

b

f =

u(z(t))x (t)dt −

a

C1

+i

b

v(z(t))y (t)dt

a b

u(z(t))y (t)dt + i

a

b

v(z(t))x (t)dt

(1)

a

while

d

f =

[u(z(λ(t))) + i v(z(λ(t)))][x (λ(t)) + i y (λ(t))]λ (t)dt.

(2)

c

C2

Expanding the integrand in (2) and analyzing the four terms separately, we find that they are exactly equal to the four corresponding terms in (1). For example

d

b

u(z(λ(t)))x (λ(t))λ (t)dt =

c

u(z(t))x (t)dt,

a

by the change-of-variable theorem for ordinary real integrals, and the proof is complete. The following proposition points out the dependence of the line integral on the direction of the curve. 4.6 Definition Suppose C is given by z(t), a ≤ t ≤ b. Then −C is defined by z(b + a − t), a ≤ t ≤ b. (Intuitively, −C is the point set of C traced in the opposite direction.) 4.7 Proposition

−C

f =

f. C

Proof

−C

f =−

b

f (z(b + a − t))˙z (b + a − t)dt.

a

Again, expanding the integral into real and imaginary parts and applying the changeof-variable theorem to each (real) integral, we find a f = f (z(t))˙z (t)dt = − f. −C

b

C

48


E XAMPLE 1 Suppose f (z) = x 2 + i y 2 (where x and y denote the real and imaginary parts of z, respectively), and consider C : z(t) = t + i t, 0 ≤ t ≤ 1. Then z˙ (t) = 1 + i and

1

f (z)dz =

0

C

1

(t 2 + i t 2 )(1 + i )dt = (1 + i )2

t 2 dt = 2i /3.

0

♦ E XAMPLE 2 Let f (z) =

x 1 y = 2 −i 2 , z x + y2 x + y2

and set C : z(t) = R cos t + i R sin t, 0 ≤ t ≤ 2π,

R = 0.

Then

2π

f (z)dz = C

0 2π

=

sin t cos t −i R R

(−R sin t + i R cos t)dt

i dt = 2πi (See Exercise 8.)

0

That is, the integral of 1/z around any circle centered at the origin (traversed counterclockwise) is 2πi . ♦ E XAMPLE 3 Suppose f (z) ≡ 1, and let C be any smooth curve. Then

f (z)dz = C

b

z˙ (t)dt = z(b) − z(a).

a

♦ The integrals defined above are natural generalizations of the definite integral and, not too surprisingly, they share many of the same properties. 4.8 Proposition Let C be a smooth curve; let f and g be continuous functions on C; and let α be any complex number. Then


I. II.

C

C

[ f (z) + g(z)]dz = α f (z)dz = α

C

49

f (z)dz +

C

C

g(z)dz

f (z)dz.

Proof

Exercise 4.

Notation: If α and β are complex numbers, the symbol α β will be used to denote the inequality |α| ≤ |β|. 4.9 Lemma Suppose G(t) is a continuous complex-valued function of t. Then

b

b

G(t)dt

a

|G(t)|dt.

a

Proof Suppose

b

G(t)dt = Reiθ , R ≥ 0.

(1)

a

By Proposition 4.8, then

b

e−iθ G(t)dt = R.

(2)

a

Suppose further that e−iθ G(t) = A(t) + i B(t), with A and B real-valued. Then, according to (2), b b R= A(t)dt = Re (e−iθ G(t))dt. a

a

But Re z ≤ |Re z| ≤ |z|, hence

b

R≤

|G(t)|dt.

(3)

a

A comparison of (1) and (3) then gives the desired result.

4.10 M-L Formula Suppose that C is a (smooth) curve of length L, that f is continuous on C, and that f M throughout C. Then f (z)dz M L. C

50


Proof Suppose C is given by z(t) = x(t)+i y(t), a ≤ t ≤ b. Then, by the previous lemma,

b

f (z)dz = a

C

b

f (z(t))˙z dt

| f (z(t))˙z (t)|dt.

a

According to the Mean-Value Theorem for Integrals applied to the positive functions | f (z(t))| and |˙z (t)|

b

f (z)dz) max | f (z)| z∈C

C

|˙z (t)|dt.

(4)

a

Finally, recall that for any curve given parametrically by (x(t), y(t)), a ≤ t ≤ b, the arc length L is given by

b

L=

(x (t))2

+ (y (t))2 dt

a

so that according to (4)

C

b

=

|˙z (t)|dt,

a

f (z)dz M L.

E XAMPLE Let C be the unit circle and suppose f 1 on C. Then M = 1, L = 2π, and f (z)dz 2π. C

To see that the upper bound of 2π can actually be achieved, consider Example 2 above. ♦ 4.11 Proposition Suppose { f n } is a sequence of continuous functions and f n → f uniformly on the smooth curve C. Then f (z)dz = lim f n (z)dz. n→∞ C

C

Proof

f (z)dz − C

f n (z)dz = C

[ f (z) − f n (z)]dz C

by Proposition 4.8. Taking n large enough so that | f (z) − f n (z)| < for all z ∈ C, and applying Proposition 4.10, shows that f (z) − fn (z)dz · (length of C) C

C


51

for any pre-assigned > 0, and hence that lim

n→∞ C

f n (z)dz =

f (z)dz. C

The following generalization of the Fundamental Theorem of Calculus will be crucial in the development of this chapter. 4.12 Proposition Suppose f is the derivative of an analytic function F–that is, f (z) = F (z), where F is analytic on the smooth curve C. Then f (z)dz = F(z(b)) − F(z(a)). C

Proof The proof depends on a complex analogue of the chain-rule for differentiation. Letting γ (t) = F(z(t)), a ≤ t ≤ b, we wish to show that γ˙ (t) = f (z(t))˙z (t) at the all-but-finite number of points where z˙ (t) exists and is nonzero. Note first that for any smooth curve λ(t), by considering the real and imaginary parts of λ separately, it is easily seen that λ(t + h) − λ(t) ˙ λ(t) = lim . h h→0 h real

Hence F(z(t + h)) − F(z(t)) h→0 h F(z(t + h)) − F(z(t)) z(t + h) − z(t) · . = lim h→0 z(t + h) − z(t) h

γ˙ (t) = lim

[Since z˙ (t) = 0, we can find δ > 0 so that |h| < δ implies z(t + h) − z(t) = 0.] Thus γ˙ (t) = f (z(t))˙z (t). Proposition 4.12 follows then by noting that b f (z)dz = f (z(t))˙z (t)dt = C

a

b

γ˙ (t)dt

a

= γ (b) − γ (a) = F(z(b)) − F(z(a)).

52


4.2 The Closed Curve Theorem for Entire Functions 4.13 Definition A curve C is closed if its initial and terminal points coincide–i.e., if C is given by z(t), a ≤ t ≤ b, with z(a) = z(b). C is a simple closed curve if no other points coincide; i.e., if z(t1 ) = z(t2 ) with t1 < t2 implies t1 = a and t2 = b. The following theorem is the first of several which show that, under rather general conditions, the integral of an analytic function along a closed curve is zero. Of course, Example 2 showed that this is not always the case. We begin cautiously, by considering entire functions. Note: Throughout the text, the boundary of a rectangle will mean a simple closed curve parametrized so that the rectangle it bounds lies on the left as the curve is traced out for increasing t. 4.14 Rectangle Theorem Suppose f is entire and is the boundary of a rectangle R. Then f (z)dz = 0.

Lemma If f is a linear function and if is as above, then f (z)dz = 0.

Proof of Lemma Let f (z) = α + βz and let be given by : z(t), a ≤ t ≤ b. Since f (z) is everywhere the derivative of the analytic function F(z) = αz + βz 2 /2, f (z)dz = F (z)dz = F(z(b)) − F(z(a)) = 0

by Proposition 4.12 and the observation that is a closed curve. (An alternate, more direct proof is outlined in Exercise 7.) Proof of Theorem 4.14 Let f (z)dz = I . To show that I = 0, we use the method of continued bisection. That is, we split the rectangle R into four congruent subrectangles, by

4.2 The Closed Curve Theorem for Entire Functions

53

bisecting each of the sides. If we let 1 , 2 , 3 , 4 denote the boundaries of the four subrectangles 4

f = f

i=1

i

Γ2

Γ1

Γ3

Γ4

since the integrals along the interior lines appear in opposite directions and thus cancel, by Proposition 4.7. Hence for some k , 1 ≤ k ≤ 4, which we will denote (1), I f (z)dz . (1) 4 Let R (1) be the rectangle bounded by (1) . Continuing in this manner, dividing into four congruent rectangles, we obtain a sequence of rectangles

R (k)

R (1) ⊃ R (2) ⊃ R (3) ⊃ · · · and their boundaries

(1) , (2) , (3) , . . .

such that diam R (k+1) = 12 diamR (k) and such that I f (z)dz k . (k) 4

(1)

(k) Let z 0 ∈ ∩∞ k=1 R . The proof will follow by considering the analyticity of f at z 0 . That is, since f (z) − f (z 0 ) → f (z 0 ) z − z0

we can write

f (z) = f (z 0 ) + f (z 0 )(z − z 0 ) + z (z − z 0 )

where z → 0 as z → z 0 . Note, then, that f (z)dz = [ f (z 0 ) + f (z 0 )(z − z 0 ) + z · (z − z 0 )]dz (n) (n) z · (z − z 0 )dz by the lemma. = (n)

54


To estimate the integral, let us assume that the largest side of the original boundary was of length s. Then, by elementary geometric considerations, 4s |dz| = length of (n) ≤ n (n) 2 √ 2·s |z − z 0 | ≤ for all z ∈ (n) . 2n Given > 0, we choose N so that √ 2·s implies that z . |z − z 0 | ≤ 2N and

Then for n ≥ N, we have by the M-L formula (Proposition 4.10) √ 4 2s 2 f (z)dz · . 4n (n)

(2)

A combination of (1) and (2) shows that for n ≥ N √ 4 2s 2 I 4n 4n or √ I · 4 2s 2 . Since this holds for all > 0, we may conclude that I = 0.

Note: Although the orientation of was chosen to be counterclockwise, the same result would hold with the opposite orientation. This follows from Proposition 4.7. The counterclockwise orientation was chosen primarily to fix a direction. In later chapters, we will see that the counterclockwise direction along the boundary is also the more “natural” one in a sense for functions analytic inside a region. Hence, unless otherwise specified, the integral around any convex curve will always be taken in the counterclockwise direction. 4.15 Integral Theorem If f is entire, then f is everywhere the derivative of an analytic function. That is, there exists an entire F such that F (z) = f (z) for all z. Proof We define F(z) as

where to z.

z 0

z

f (ζ )dζ

0

denotes the integral along the straight lines from 0 to Re z and from Re z

4.2 The Closed Curve Theorem for Entire Functions

Note that

55

z+h

F(z + h) = F(z) +

f (ζ )dζ

z

where the integral denotes the integral along the line segments from z to z + Re h and from z + Re h to z + h. This follows since the difference between the two approaches is equal to the integral of f around a closed rectangle and is thus equal to zero. (See diagrams below.) z+h z

z+h

z + Re h

z + Re h

0

z

0

Hence

z+h

F(z + h) − F(z) =

f (ζ )dζ

z

and since 1 h

z+h

1dz =

z

1 (z + h − z) = 1, h

(see Example 3 after Proposition 4.7) 1 F(z + h) − F(z) − f (z) = h h

z+h

[ f (ζ ) − f (z)]dζ.

z

Finally, for each ≥ 0, if h is small enough, | f (ζ ) − f (z)| throughout the path of integration. Applying the M-L formula, we obtain 1 F(z + h) − F(z) − f (z) · 2h = 2. h h Hence

F (z) = f (z).

56


4.16 Closed Curve Theorem If f is entire and if C is a (smooth) closed curve, f (z)dz = 0. C

Proof Since f is entire, by the Integral Theorem f (z) = F (z) for some entire function F so that f (z)dz = F (z)dz = F(z(b)) − F(z(a)) C

C

by Proposition 4.12. Since C is closed, z(b) = z(a), F(z(b)) = F(z(a)), and f (z)dz = 0. C

Remarks While Theorem 4.16 was proven for entire functions f , the only fact we needed was that f (z) is the derivative of an analytic function on C. Thus, for example, 1 dz = 0 2 C z if C is any smooth closed curve not passing through the origin. For although 1/z 2 is not entire, it is the derivative of F(z) = −1/z which is analytic except at the origin. Similarly, z k dz = 0 C

if k is any integer except −1. Recall Example 2 which showed that k = −1 is an exception to the above. (See Exercise 8.)

Exercises 1. Prove that “equivalence” of smooth curves has the familiar reflexive, symmetric, and transitive properties of an equivalence relation. 2. Evaluate C f where f (z) = x 2 + iy 2 as in Example 1, but where C is given by z(t) = t 2 + it 2 , 0 ≤ t ≤ 1. 3. Evaluate C f where f (z) = 1/z as in Example 2, and C is given by z(t) = sin t +i cos t, 0 ≤ t ≤ 2π . Why is the result different from that of Example 2? 4. Prove Proposition 4.8. [Hint: Divide the integrals into real and imaginary parts.] 5. Prove the uniqueness of the integral. That is, show that F ≡ 0 implies that F is a constant. [Hint: Use Proposition 4.12 to get an expression for F(b) − F(a).]

Exercises

57

6. Show that, if f is a continuous real-valued function and f 1, then f 4.

|z|=1

[Hint: Show that

f 02π | sin t|dt.]

7. Give a direct proof of the lemma to Theorem 4.14. That is, given any rectangle with vertices (a, c), (b, c), (b, d) and (a, d), parameterize the boundary and verify directly that

dz =

zdz = 0.

8. Show that C z k dz = 0 for any integer k = −1 and C : z = Re iθ , 0 ≤ θ ≤ 2π a. by showing that z k is the derivative of a function analytic throughout C, b. directly, using the parametrization of C. 9. Evaluate C (z − i)dz where C is the parabolic segment: z(t) = t + it 2 , −1 ≤ t ≤ 1 a. by applying Proposition 4.12, b. by integrating along the straight line from −1 + i to 1 + i and applying the Closed Curve Theorem. 10.* Evaluate a. 0i ez dz π/2+i b. π/2 cos 2z dz 11.* Suppose f is analytic in a convex region D and | f | ≤ 1 throughout D. Prove that f is a "contraction"; i.e., show that | f (b) − f (a)| ≤ |b − a| for all a, b ∈ D. 12.* Let a, b be two complex numbers in the left half-plane. Prove that |ea − eb | < |a − b|.

Chapter 5

Properties of Entire Functions

5.1 The Cauchy Integral Formula and Taylor Expansion for Entire Functions We now show that if f is entire and if ⎧ ⎪ ⎨ f (z) − f (a) z−a g(z) = ⎪ ⎩ f (a)

z = a z=a

then the Integral Theorem (4.15) and Closed Curve Theorem (4.16) apply to g as well as to f . (Note that since f is entire, g is continuous; however, it is not obvious that g is entire.) We begin by showing that the Rectangle Theorem applies to g. 5.1 Rectangle Theorem II If f is entire and if ⎧ ⎪ ⎨ f (z) − f (a) z−a g(z) = ⎪ ⎩ f (a) then

z = a z=a

g(z)dz = 0, where is the boundary of a rectangle R.

Proof We consider three cases. I. a ∈ ext R. In this case, g is analytic throughout R and the proof is exactly the same as that of Theorem 4.14. Note that the proof required only that the integrand be analytic throughout R and .

59

60

5 Properties of Entire Functions

II. a ∈ . Divide R into six subrectangles as indicated and note that because of the cancellations involved 6

g= g (1)

k=1 k

where k , 1 ≤ k ≤ 6, denote the boundaries of the subrectangles. Since g is ¯ g M for some constant M. If we take continuous in the compact domain R, the boundary of the rectangle containing a (call it 1 ) to have length less than , g M by the M-L formula 1

while

k

as in case (I). Hence by (1)

g = 0,

k = 1

g M

for any > 0 and the proof is complete. III. a ∈ int R.

a

5.1 The Cauchy Integral Formula and Taylor Expansion for Entire Functions

61

Here, as in the previous case, we subdivide R–this time into nine rectangles. Along the boundaries of the eight rectangles (not containing a) g = 0, k

while the integral along the boundary of the remaining subrectangle can be made arbitrarily small by choosing its length to be as small as required. As in the previous case, we conclude

g=

9

k=1 k

g = 0.

5.2 Corollary Suppose g is as above. Then the Integral Theorem and the Closed Curve Theorem apply to g. Proof We observe that since g is continuous, the proofs of Theorems 4.15 and 4.16 apply, without any modification, to g. 5.3 Cauchy Integral Formula Suppose that f is entire, that a is some complex number, and that C is the curve C : Reiθ ,

0 ≤ θ ≤ 2π,

Then f (a) =

1 2πi

C

with R > |a|. f (z) dz. z−a

Proof By Corollary 5.2

f (z) − f (a) dz = 0 z−a

C

so that

f (a) C

dz = z−a

C

f (z) dz z−a

and the proof follows once we show that dz = 2πi. C z−a This lemma is proven below in somewhat greater generality.

62


5.4 Lemma Suppose a is contained in the circle Cρ : that is, Cρ has center α, radius ρ, and |a − α| < ρ. Then dz = 2πi. Cρ z − a Proof First we note that

Cρ

while

Cρ

dz = z−α

2π 0

dz =0 (z − α)k+1

iρeiθ dθ = 2πi, ρeiθ for k = 1, 2, 3, . . . .

The second equality follows not only from a direct evaluation of the integral Cρ

i dz = k (z − α)k+1 ρ

2π

e−ikθ dθ = 0

0

but also from the fact that 1/(z − α)k+1 is equal to the derivative of the analytic k function −1/k(z − α) . To evaluate Cρ (1/(z − a))dz, write 1 1 1 = = z−a (z − α) − (a − α) (z − α)[1 − (a − α)/(z − α)] 1 1 = · (z − α) 1 − ω where ω=

a−α |a − α| has fixed modulus < 1 throughout Cρ . z−α ρ

(1)

By (1) and the fact that 1/(1 − ω) = 1 + ω + ω2 + · · · , we obtain 1 a−α (a − α)2 1 = + 1+ + ··· z−a z−α z−α (z − α)2 =

a−α 1 (a − α)2 + + + ··· . z−α (z − α)2 (z − α)3

Since the convergence is uniform throughout Cρ , Cρ

1 dz = z−a

∞

Cρ

1 dz + z−α

k=1 C ρ

(a − α)k dz = 2πi. (z − α)k+1

5.1 The Cauchy Integral Formula and Taylor Expansion for Entire Functions

63

5.5 Taylor Expansion of an Entire Function If f is entire, it has a power series representation. In fact, f (k) (0) exists for k = 1, 2, 3, . . . , and ∞

f (k) (0) k z f (z) = k! k=0

for all z. Proof Suppose a = 0, R = |a| + 1 and let C be the circle: |ω| = R. By the Cauchy Integral Formula f (ω) 1 dω f (z) = 2πi C w − z for all z a. As before, note that 1 1 z 1 z2 = = + + + ··· , z ω−z ω ω2 ω3 ω 1− ω and since the convergence is uniform throughout C z 1 z2 1 + f (z) = f (ω) + 3 + · · · dω 2πi C ω ω2 ω f (ω) f (ω) f (ω) 1 1 1 dω + dω z + dω z2 + · · · = 2πi C ω 2πi C ω2 2πi C ω3 ∞

Ck z k = k=0

where Ck =

1 2πi

C

f (ω) dω. ωk+1

(1)

Since for each z, there exists some a z, the proof of the first part of the theorem appears to be complete. There is, however, one wrinkle. The contour C–and hence the coefficients of the power series–depended on a, for the radius R had to be chosen larger than |a| to insure the uniform convergence of the power series for 1/(ω − z). On the other hand, if we think of a as being fixed, we have shown that there exist coefficients C0 (a), C1 (a), C2 (a), . . . , such that

(2) f (z) = Ck (a)z k for all z a. To see that this is sufficient we note that although, a priori, the coefficients could change as we consider complex numbers a of increasing magnitude, they are in fact constant.

64


For, as we saw in Chapter 2 (Corollary 2.11), it follows from (2) that f is infinitely differentiable at 0 and that f (k) (0) . Ck (a) = k! Hence the coefficients are independent of a. Note, finally, that although the everywhere convergence of the series

f (k) (0) zk k! is not proven explicitly, it is implicit in the fact that the series equals f (z) for all z. 5.6 Corollary An entire function is infinitely differentiable. Proof Since f has a power series expansion, we may invoke Corollary 2.10–an everywhere convergent power series is infinitely differentiable. 5.7 Corollary If f is entire and if a is any complex number, then f (z) = f (a) + f (a)(z − a) +

f (a) (z − a)2 + · · · 2!

for all z.

Proof Consider g(ζ ) = f (ζ + a) which is likewise entire. By 5.5 g(ζ ) = g(0) + g (0)ζ +

g (0) 2 ζ + ··· , 2!

so that f (ζ + a) = f (a) + f (a)ζ +

f (a) 2 ζ + ··· . 2!

Setting ζ = z − a, the corollary follows.

5.8 Proposition If f is entire and if ⎧ ⎨ f (z) − f (a) z−a g(z) = ⎩ f (a) then g is entire.

z = a z=a

5.2 Liouville Theorems and the Fundamental Theorem of Algebra

65

Proof By the previous corollary, for z = a f (3) (a) f (a) (z − a) + (z − a)2 + · · · , (1) 2! 3! and by the definition of g, (1) is also valid at z = a. Since g is equal to an everywhere convergent power series, g is entire. g(z) = f (a) +

5.9 Corollary Suppose f is entire with zeroes at a1 , a2 , . . . , a N . Then if g is defined by g(z) =

f (z) for z = ak , (z − a1 )(z − a2 ) . . . (z − a N )

limz→ak g(z) exists for k = 1, 2, . . . , N, and if g(ak ) is defined by these limits, then g is entire. Proof Let f0 (z) = f (z) and let f k (z) =

f k−1 (z) − f k−1 (ak ) f k−1 (z) = , z − ak z − ak

z = ak .

Assuming that f k−1 is entire, it follows from Proposition 5.8 that fk (z) has a limit as z → ak and if we define f k (ak ) to be this limit, f k is entire. Since f 0 is entire by hypothesis, the proof follows by induction.

5.2 Liouville Theorems and the Fundamental Theorem of Algebra; The Gauss-Lucas Theorem 5.10 Liouville’s Theorem A bounded entire function is constant. Proof Let a and b represent any two complex numbers and let C be any positively oriented circle centered at 0 and with radius R > max(|a|, |b|). Then according to the

66


Cauchy Integral Formula (5.3) f (b) − f (a) =

1 2πi

C

1 f (z) dz − z−b 2πi

C

f (z) dz z−a

1 f (z)(b − a) dz 2πi C (z − a)(z − b) M|b − a| · R (R − |a|)(R − |b|)

=

(1)

using the usual estimate, where M represents the supposed upper bound for | f |. Since R may be taken as large as desired and since the expression in (1) approaches 0 as R → ∞, f (b) = f (a) and f is constant. 5.11 The Extended Liouville Theorem If f is entire and if, for some integer k ≥ 0, there exist positive constants A and B such that | f (z)| ≤ A + B|z|k , then f is a polynomial of degree at most k. Proof Note that the case k = 0 is the original Liouville Theorem. The general case follows by induction. Thus, we consider ⎧ ⎪ ⎨ f (z) − f (0) z = 0 z g(z) = ⎪ ⎩ f (0) z = 0. By 5.8, g is entire and by the hypothesis on f , |g(z)| ≤ C + D|z|k−1 . Hence g is a polynomial of degree at most k − 1 and f is a polynomial of degree at most k. 5.12 Fundamental Theorem of Algebra Every non-constant polynomial with complex coefficients has a zero in C. Proof Let P(z) be any polynomial. If P(z) = 0 for all z ∈ C, f (z) = 1/P(z) is an entire function. Furthermore if P is non-constant, P → ∞ as z → ∞ and f is bounded. But then, by Liouville’s Theorem, f is constant, and so is P, contrary to our assumption.

5.2 Liouville Theorems and the Fundamental Theorem of Algebra

67

Remarks 1. If α is a zero of an n-th degree polynomial Pn , Pn (z) = (z − α)Pn−1 (z), where Pn−1 is a polynomial of degree n − 1. This can be seen by the usual Euclidean Algorithm or by noting that Pn (z) n−1 z − α ≤ A + B|z| and hence is equal to an (n − 1)-st degree polynomial by the Extended Liouville Theorem. 2. α is called a zero of multiplicity k (or order k) if P(z) = (z − α)k Q(z), where Q is a polynomial with Q(α) = 0. Equivalently, α is a zero of multiplicity k if P(α) = P (α) = · · · = P (k−1) (α) = 0, P (k) (α) = 0. The equivalence of the two definitions is easily established and is left as an exercise. 3. Although the Fundamental Theorem of Algebra only assures the existence of a single zero, an induction argument shows that an n-th degree polynomial has n zeroes (counting multiplicity). For, assuming every k-th degree polynomial can be written Pk (z) = A(z − z 1 ) · · · (z − z k ), it follows that Pk+1 (z) = A(z − z 0 )(z − z 1 ) · · · (z − z k ). By the above remark, any polynomial Pn (z) = an z n + an−1 z n−1 + · · · + a0

(2)

can also be expressed as Pn (z) = an (z − z 1 )(z − z 2 ) · · · (z − z n ),

(3)

where z 1 , z 2 , ...z n are the zeroes of Pn . A comparison of (2) and (3) yields the well-known relations between the zeroes of a polynomial and its coefficients. For example,

z k = −an−1 /an . (4) There are many entire functions, such as e z − 1, which have infinitely many zeroes, and whose derivatives are never zero. So there is no general analytic analogue of Rolle’s Theorem. However, for polynomials, the Gauss-Lucas Theorem, below, offers a striking analogy and, in some ways a stronger form, of Rolle’s Theorem. Recall that a convex set is one that contains the entire line segment connecting any two of its points. Hence, if z 1 and z 2 belong to a convex set, so does every complex number of the form tz 1 + (1 − t)z 2 , for 0 ≤ t ≤ 1. We leave it as an exercise to show that if z 1 , z 2 , ..., z n belong to a convex set, so does every “convex” combination of the form

a1 z 1 + a2 z 2 + · · · + an z n ; ai ≥ 0 for all i , and ai = 1. (5)

68


5.13 Definition The convex hull of a set S of complex numbers is the smallest convex set containing S. 5.14 Gauss-Lucas Theorem The zeroes of the derivative of any polynomial lie within the convex hull of the zeroes of the polynomial. Proof Assume that the zeroes of P are z 1 , z 2 , ..., z n and that α is a zero of P but not a zero of P, Then 1 1 1 P (α) = + +···+ =0 (6) P(α) α − z1 α − z2 α − zn Rewriting

1 α − zi = α − zi |α − z i |2

we can apply (6) to obtain α=

1 ai z i , with ai = |α − z i |2

1 . |α − z i |2

(7)

Finally, by taking conjugates in (7), we obtain an identical expression for α in terms of z 1 , z 2 , ..., z n . Hence α is in the convex hull of {z 1 , z 2 , ..., z n }. A final remark The Fundamental Theorem of Algebra can be considered a “nonexistence theorem” in the following sense. Recall that the complex numbers come into consideration when the reals are supplemented to include a solution of the equation x 2 + 1 = 0. One might have supposed that further extensions would arise as we sought zeroes of other polynomials with real or complex coefficients. By the Fundamental Theorem of Algebra, all such solutions are already contained in the field of complex numbers, and hence no such further extensions are possible. This is usually expressed by saying that the field of complex numbers is algebraically closed.

5.3 Newton’s Method and Its Application to Polynomial Equations I. Introduction We saw in Chapter 1 that solutions of quadratic and cubic equations can be found in terms of square roots and cube roots of various expressions involving the coefficients. A similar formula is also available for fourth degree polynomial equations. On the other hand, one of the highlights of modern mathematics is

5.3 Newton’s Method and Its Application to Polynomial Equations

69

the famous theorem that no such solution, in terms of n-th roots, can be given for the general polynomial equation of degree five or higher. In spite of this, there are many graphing calculators that allow the user to input the coefficients of a polynomial of any degree and then almost immediately output all of its zeroes, correct to eight or nine decimal places. The explanation for this magic is that, although there are no formulas for solving all polynomial equations, there are many algorithms which can be used to find arbitrarily good approximations to the solutions. One extremely popular and effective method for approximating solutions to equations of the form f (z) = 0, variations of which are incorporated in many calculators, is known as Newton’s Method. It can be informally described as follows: i) Choose a point z 0 “sufficiently close” to a solution of the equation, which we will call s. ii) Define z 1 = z 0 − f (z 0 )/ f (z 0 ) and continue recursively, defining z n+1 = z n − f (z n )/ f (z n ). Then, if z 0 is sufficiently close to the root s, the sequence {z n } will converge to s. In fact, the convergence is usually extremely rapid. If we are trying to approximate a real solution s to the “real” equation f (x) = 0, the algorithm has a very nice geometric interpretation. That is, suppose (x 0 , f (x 0 )) is a point P on the graph of the function y = f (x).Then the tangent to the graph at point P is given by the equation L(x) = f (x 0 ) + f (x 0 )(x − x 0 ). Hence x 1 = x 0 − f (x 0 )/ f (x 0 ) is precisely the point where the tangent line crosses the x-axis.

y xk+1 = xk —

f (xk) f ' (xk)

y = f (x)

xk

xk+1

x

Similarly, x n+1 is the zero of the tangent to y = f (x) at the point (x n , f (x n )). Thus, there is a very clear visual insight into the nature of the sequence generated by the algorithm and it is easy to convince oneself that the sequence converges to the solution s in most cases. However, the geometric argument leaves many questions unanswered. For example, how do we know if x 0 is sufficiently close to the root s? Furthermore, if the sequence does converge, how quickly does it converge? Experimenting with simple examples will verify the assertion made earlier that the convergence is, in fact, very quick, but why is it? Finally, and of special interest to us,

70


why does the method work in the complex plane, where the geometric interpretation is no longer applicable? The answer to all these questions can be found by taking a slight detour into the topic of fixed-point iteration. II. Fixed-Point Iteration Suppose we are given an equation in the form z = g(z). Then a solution s is a “fixed-point” of the function g. As we will see below, under the proper conditions, approximating such a fixed point can often be accomplished by recursively defining z n+1 = g(z n ), a process known as fixed point iteration. 5.15 Lemma Let s denote a root of the equation z = g(z), for some analytic function g. Suppose that z 0 belongs to a disc of the form D(s; r ) throughout which |g (z)| ≤ K , and let z 1 = g(z 0 ). Then |z 1 − s| ≤ K |z 0 − s|. Proof Note that |z 1 − s| = |g(z 0 ) − g(s)|. Using the complex version of the Fundamental Theorem of Calculus, z0 g(z 0 ) − g(s) = g (z)dz s

where we choose the path of integration to be the straight line from s to z 0 . The result then follows immediately from the M − L formula. 5.16 Theorem Let s denote a root of the equation z = g(z),for some analytic function g. Suppose that z 0 belongs to a disc of the form D(s; r ) throughout which |g (z)| ≤ K < 1 and define the sequence {z n } recursively as: z n+1 = g(z n ); n = 0, 1, 2, .... Then {z n } → s as n → ∞. Proof Note that, as in Lemma 5.15, |z n+1 − s| ≤ K |z n − s| and hence, by induction, z n ∈ D(s; r ) for all n and |z n − s| ≤ K n |z 0 − s|. Since K < 1, the result follows immediately. 5.17 Corollary Let s denote a root of the equation z = g(z), for some analytic function g and assume that |g (s)| < 1. Then there exists a disc of the form D(s; r ) such that if z 0 ∈ D(s:r )


71

and if we define the sequence {z n } recursively as: z n+1 = g(z n ); n = 0, 1, 2, ...., {z n } → s as n → ∞. Proof Since |g (s)| < 1, there exists a constant K with |g (s)| < K < 1. But then, since g is analytic, there must exist exist a disc D(s; r ) throughout which |g (z)| < K . Suppose we let εn = |z n − s| denote the n-th error, i.e. the absolute value of the difference between the n-th approximation z n and the desired solution, s. Then the above results show that, with an appropriate starting value z 0 , the sequence of errors satisfies the inequality εn+1 ≤ K εn (1) 1 for every 3 or 4 iterations. If, e.g. K = 12 , the error will be reduced by a factor of 10 An iteration scheme which satisfies inequality (1) for any value of K , 0 < K < 1, is said to converge linearly. In that case, the number of iterations required to obtain n decimal place accuracy is roughly proportional to n. Corollary 5.17 shows that an important condition for the convergence of fixedpoint iteration is that |g (s)| < 1 This raises the following practical problem. An equation in the familiar form f (z) = 0 can certainly be rewritten as an equivalent equation in the fixed point form z = g(z). For example, one could simply add the monomial z to both sides of the equation. But how can we rewrite f (z) = 0 in the form z = g(z) with the additional condition that |g (s)| < 1 at the unknown solution s ? One answer to this problem will provide the insight to Newton’s method that we are looking for. That is, suppose the equation f (z) = 0 is rewritten in the form z = g(z) = z − f (z)/ f (z). Then the fixed point iteration algorithm is precisely Newton’s Method. Moreover, we can find the exact value of g (s)!!

5.18 Lemma If f is analytic and has a zero of order k at z = s, and if g(z) = z − f (z)/ f (z), then g is also analytic at s and g (s) = 1− 1k . Proof By hypothesis, f (z) = (z − s)k h(z), with h(s) = 0. Hence f (z)/ f (z) =

(z − s)h(z) kh(z) + (z − s)h (z)

Thus f / f is analytic at s (with the appropriate value of 0 at s), and its power series expansion about the point s is of the form 1k (z − s) + a2 (z − s)2 + · · ·. Hence g (s) = 1− 1k

72


Applying Corollary 5.17 then yields 5.19 Theorem Let s denote a root of the equation f (z) = 0. Let g(z) = z − f (z)/ f (z), and define the sequence {z n } recursively as: z n+1 = g(z n ); n = 0, 1, 2, .. Then there exists a disc of the form D(s; r ) such that z 0 ∈ D(s; r ) guarantees that {z n } → s as n → ∞. If f (z) has a simple zero at s, according to Lemma 5.18, g(z) = z − f (z)/ f (z) will have g (s) = 0. In this case, the iteration scheme will converge especially rapidly. 5.20 Lemma Let s denote a root of the equation z = g(z), for some analytic function g such that g (s) = 0. Suppose that z 0 belongs to a disc of the form D(s; r ) throughout which |g (z)| ≤ M and let z 1 = g(z 0 ). Then |z 1 − s| ≤ 12 M|z 0 − s|2 . Proof As in lemma 5.15, we begin by noting that z 1 − s = g(z 0 ) − g(s) = But for any value of z on the line segment [s, z 0 ], we can write: z |g (z)| = |g (z) − g (s)| = | g (z)dz| ≤ M|z − s|

z0 s

g (z)dz.

(2)

s

Let z = (z 0 − s)/n and write s+z z0 g (z)dz = g + s

s

s+2z

g + .... +

s+z

z0 z 0 −z

g

(3)

Then applying the M-L formula z to each of the integrals in (3) and using the estimates for g given by (2) show that s 0 g (z)dz is bounded by n

k=1

Mk(z)2 = M

n(n + 1) |z 0 − s|2 2 n2

and the lemma follows by letting n → ∞.

5.21 Definition If εn = |z n − s| satisfies εn+1 ≤ K εn2 , we say that the sequence {z n } converges quadratically to s. Note that in the case of quadratic convergence, once the sequence of iterations is close to its limit, each iteration virtually doubles the number of decimal places which are accurate. If, for example, at some point the error is in the 10th decimal


73

place, then at that point, εn is approximately 10−10 , so that εn+1 = K εn2 will be approximately 10−20 . Lemmas 5.18 and 5.20 combine then to give us 5.22 Theorem If f (z) has a simple zero at a point s, and if z 0 is sufficiently close to s, Newton’s Method will produce a sequence which converges quadratically to s. III. Newton’s Method Applied to Polynomial Equations While Newton’s Method can be (and is) applied to all sorts of equations, it works especially well for polynomial equations. For one thing, we don’t have to worry about the existence of solutions; they are guaranteed by the Fundamental Theorem of Algebra. That may be one reason why Newton himself applied his method only to polynomial equations. According to Theorems 5.19 and 5.22, as long as the initial approximation z 0 is sufficiently close to one of the roots, Newton’s Method will converge to it. If we are looking for a simple zero of a polynomial, the method will actually converge quadratically. Of course, there are starting points which will not yield a convergent sequence. For example, if z 0 is a zero of the derivative of the polynomial, z 1 will not be defined! On the other hand, the set of “successful” starting points is surprisingly robust. Modern technology has been applied to identifying what have been labeled “Newton basins”, the distinct regions in the complex plane from which a starting value will yield a sequence converging to the distinct zeroes of a polynomial. If these regions are shaded in different colors, they yield remarkably interesting sketches. Aside from the example below, interested readers can generate their own sketches of the Newton basins for various polynomials at http://aleph0.clarku.edu/∼djoyce/newton/ technical.html The sketch below shows the Newton basins for the eight zeroes of the polynomial P(z) = (z 4 − 1)(z 4 + 4). The eight roots: ±1, ±i, ±(1 + i ), ±(1 − i ) are at the corners and the midpoints of the sides of the displayed square. The black regions contain the starting points which do not yield a convergent sequence.

74


Exercises 1. Find the power series expansion of f (z) = z 2 around z = 2. 2. Find the power series expansion for ez about any point a. 3. f is called an odd function if f (z) = − f (−z) for all z; f is called even if f (z) = f (−z). a. Show that an odd entire function has only odd terms in its power series expansion about z = 0. [Hint: show f odd ⇒ f even, etc., or use the identity f (z) =

f (z) − f (−z) .] 2

b. Prove an analogous result for even functions. 4. By comparing the different expressions for the power series expansion of an entire function f , prove that k! f (ω) dω, k = 0, 1, 2, . . . f (k) (0) = 2π i C ωk+1 for any circle C surrounding the origin. 5. (A Generalization of the Cauchy Integral Formula). Show that f (k) (a) =

k! f (ω) dω, 2π i C (ω − a)k+1

k = 1, 2, . . .

where C surrounds the point a and f is entire. 6. a. Suppose an entire function f is bounded by M along |z| = R. Show that the coefficients Ck in its power series expansion about 0 satisfy M . Rk b. Suppose a polynomial is bounded by 1 in the unit disc. Show that all its coefficients are bounded by 1. |Ck | ≤

7. (An alternate proof of Liouville’s Theorem). Suppose that | f (z)| ≤ A + B|z|k and that f is entire. Show then that all the coefficients C j , j > k, in its power series expansion are 0. (See Exercise 6a.) 8. Suppose f is entire and | f (z)| ≤ A + B|z|3/2. Show that f is a linear polynomial. 9. Suppose f is entire and | f (z)| ≤ |z| for all z. Show that f (z) = a + bz 2 with |b| ≤ 12 . 10. Prove that a nonconstant entire function cannot satisfy the two equations i. f (z + 1) = f (z) ii. f (z + i) = f (z) for all z. [Hint: Show that a function satisfying both equalities would be bounded.] 11. A real polynomial is a polynomial whose coefficients are all real. Prove that a real polynomial of odd degree must have a real zero. (See Exercise 5 of Chapter 1.) 12. Show that every real polynomial is equal to a product of real linear and quadratic polynomials. 13. Suppose P is a polynomial such that P(z) is real if and only if z is real. Prove that P is linear. [Hint: Set P = u + iv, z = x + iy and note that v = 0 if and only if y = 0. Conclude that: a. either v y ≥ 0 throughout the real axis or v y ≤ 0 throughout the real axis; b. either u x ≥ 0 or u x ≤ 0 for all real values and hence u is monotonic along the real-axis; c. P(z) = α has only one solution for real-valued α.]

Exercises

75

14. Show that α is a zero of multiplicity k if and only if P(α) = P (α) = · · · = P (k−1) (α) = 0, and P (k) (α) = 0. 15. Suppose that f is entire and that for each z, either | f (z)| ≤ 1 or | f (z)| ≤ 1. Prove that f is a linear polynomial. [Hint: Use a line integral to show | f (z)| ≤ A + |z| where A = max(1, | f (0)|).] 16.* Let (z 1 + z 2 + · · · + z n )/n denote the centroid of the complex numbers z 1 , z 2 , ..., z n . Use formula (4) in section 5.2 to show that the centroid of the zeroes of a polynomial is the same as the centroid of the zeroes of its derivative. 17.* Use induction to show that if z 1 , z 2 , ..., z n belong to a convex set, so does every "convex" combination of the form a1 z 1 + a2 z 2 + · · · + an z n ; ai ≥ 0 for all i, and

ai = 1.

18.* Let Pk (z) = 1 + z + z 2 /2! + · · · + z k /k!, the kth partial sum of ez . a. Show that, for all values of k ≥ 1,the centroid of the zeroes of Pk is −1. b. Let z k be a zero of Pk with maximal possible absolute value. Prove that {|z k |} is an increasing sequence. 19.* Let P(z) = 1 + 2z + 3z 2 + · · · + nz n−1 . Use the Gauss-Lucas theorem to show that all the zeroes of P(z) are inside the unit disc. (See exercise 20 of Chapter 1 for a more direct proof.) √ 20.* Find estimates for i by applying Newton’s method to the polynomial equation z 2 = i, with z 0 = 1.

Chapter 6

Properties of Analytic Functions

Introduction In the last two chapters, we studied the connection between everywhere convergent power series and entire functions. We now turn our attention to the more general relationship between power series and analytic functions. According to Theorem 2.9 every power series represents an analytic function inside its circle of convergence. Our first goal is the converse of this theorem: we will show that a function analytic in a disc can be represented there by a power series. We then turn to the question of analytic functions in arbitrary open sets and the local behavior of such functions.

6.1 The Power Series Representation for Functions Analytic in a Disc 6.1 Theorem Suppose f is analytic in D = D(α; r ). If the closed rectangle R and the point a are both contained in D and represents the boundary of R, f (z) − f (a) dz = 0. f (z)dz = z−a Proof The proof is exactly the same as those of Theorems 4.14 and 5.1. The only requirement there was that f be analytic throughout R, and this is satisfied since R ⊂ D. To simplify notation, we adopt the following convention. If f (z) is analytic in a region D, including the point α, the function g(z) =

f (z) − f (a) z−a 77

78

6 Properties of Analytic Functions

will denote the function given by ⎧ ⎨ f (z) − f (a) z−a g(z) = ⎩ f (a)

z ∈ D, z = a z = a.

The fact that g is analytic at a is proven in Proposition 6.7. (Compare with Proposition 5.8.) 6.2 Theorem If f is analytic in D(α; r ), and a ∈ D(α; r ), there exist functions F and G, analytic in D and such that F (z) = f (z),

G (z) =

f (z) − f (a) . z−a

Proof

We define F(z) =

and G(z) =

z α

z α

f (ζ )dζ

f (ζ ) − f (a) dζ ζ −a

where the path of integration consists of the horizontal and then vertical segments from α to z. Note that for any z ∈ D(α; r ) and h small enough, z + h ∈ D(α; r ) so that, as in 4.15, we may apply the Rectangle Theorem to the respective difference quotients to conclude F (z) = f (z) and G (z) =

f (z) − f (a) . z−a

6.3 Theorem If f and a are as above and C is any (smooth) closed curve contained in D(α; r ), f (z) − f (a) dz = 0. f (z)dz = z−a C C Proof According to Theorem 6.2, there exists G, analytic in D(α; r ) and such that G (z) =

f (z) − f (a) . z−a

6.1 The Power Series Representation for Functions Analytic in a Disc

79

Hence, C

f (z) − f (a) dz = z−a

G (z)dz = G(z(b)) − G(z(a)) = 0 C

since the initial and terminal points z(a) and z(b) coincide. Similarly, f (z)dz = 0. C 6.4 Cauchy Integral Formula Suppose f is analytic in D(α; r ), 0 < ρ < r , and |a − α| < ρ. Then 1 f (z) dz f (a) = 2πi Cρ z − a where Cρ represents the circle α + ρeiθ , 0 ≤ θ ≤ 2π.

Cρ a

r

α

ρ

Proof Cρ

so that f (a)

Cρ

f (z) − f (a) dz = 0 z−a dz = z−a

Moreover, according to Lemma 5.4, Cρ

and the proof is complete.

Cρ

f (z) dz. z−a

dz = 2πi z−a

80


6.5 Power Series Representation for Functions Analytic in a Disc If f is analytic in D(α; r ) there exist constants Ck such that f (z) =

∞

Ck (z − α)k

k=0

for all z ∈ D(α; r ). Proof Pick a ∈ D(α; r ) and ρ > 0 such that |a − α| < ρ < r . By the previous integral formula, if |z − α| < |a − α| 1 f (ω) f (z) = dω 2πi Cρ ω − z and using the fact that z−α 1 (z − α)2 + + + ··· ω−α (ω − α)2 (ω − α)3 converges uniformly to 1/(ω − z) throughout Cρ (see Lemma 5.4) z−α (z − α)2 1 1 + f (ω) + + · · · dω f (z) = 2πi Cρ ω−α (ω − α)2 (ω − α)3 = C0 (ρ) + C1 (ρ)(z − α) + C2 (ρ)(z − α)2 + · · · where Ck (ρ) =

1 2πi

(1)

f (ω) dω. (ω − α)k+1

Cρ

Note, then, that the coefficients Ck (ρ) are actually independent of ρ. For once again, as in 5.5, we can apply (1) to conclude that f is infinitely differentiable at α and Ck (ρ) =

f (k) (α) for each ρ, 0 < ρ < r, and all k. k!

Hence, for all z ∈ D(α; r ) f (z) =

∞

Ck (z − α)k

k=0

with Ck =

1 f (k) (α) = k! 2πi

Cρ

f (z) dz. (z − α)k+1

6.2 Analytic in an Arbitrary Open Set

81

6.2 Analytic in an Arbitrary Open Set The methods used above cannot be generalized to find a single power series equal to a given analytic function in an arbitrary open set. In fact no such generalization is possible even to the most elementary of domains–e.g., a square. The breakdown in the previous strategy arises when, given a point a in the square, we try to find a contour C surrounding a and the center α of the square such that a −α ω − α < 1 for all ω ∈ C (see the diagram). As we shall soon see, this is not simply a technical difficulty but a reflection of the fact that in general, no such power series exists! However, we can apply our previous results to obtain the following general theorem.

a

α

6.6 Theorem If f is analytic in an arbitrary open domain D, then for each α ∈ D, there exist constants Ck such that ∞

f (z) = Ck (z − α)k k=0

for all points z inside the largest disc centered at α and contained in D. Proof This is a simple reformulation of Theorem 6.5.

E XAMPLES i. f (z) = 1/(z − 1) is analytic at z = 2 and in a disc of radius 1 centered at z = 2. To find a power series representation for f in that disc, we write 1 1 = = 1 − (z − 2) + (z − 2)3 − (z − 2)3 + − · · · z−1 1 + (z − 2) which converges as long as |z − 2| < 1.

(1)

82


Note that the power series diverges throughout |z − 2| > 1 despite the fact that f (z) = 1/(z − 1) is analytic everywhere except at the single point z = 1. Furthermore, according to Theorem 2.14 any other power series ak (z − 2)k which equals 1/(z − 1) in any disc around z = 2 would have with to be identical the power series in (1). Hence, there is no power series ak (z − 2)k equal to 1/(z − 1) throughout its domain of analyticity. ii. To find a power series representation for 1/z 2 near z = 3, we set 2 2 1 1 1 1 = = 3 + (z − 3) 9 1 + (z − 3)/3 z2 2 (z − 3) (z − 3)2 (z − 3)3 1 1− + − + −··· = 9 3 9 27 2(z − 3) 3(z − 3)2 4(z − 3)3 1 1− + − + −··· = 9 3 9 27 =

∞

(−1)k (k + 1) (z − 3)k . 9 3k k=0

Note again that the radius of convergence

1/ lim sup |Ck |

1/k

9 · 3k = lim k+1

1/k =3

represents the radius of the largest disc centered at z = 3 in which 1/z 2 is analytic. iii. To find the first three terms of the power series for f (z) = sin(1/z) around z = 1, because no immediate formula suggests itself, we evaluate the coefficients directly using the formula Ck =

f (k) (1) . k!

Thus we find f (z) = sin

(2 cos 1 − sin 1) 1 = sin 1 − cos 1(z − 1) + (z − 1)2 + · · · z 2

♦

6.3 The Uniqueness, Mean-Value, and Maximum-Modulus Theorems; Critical Points and Saddle Points We now consider some of the implications of the power series representations discussed in Theorem 6.6. We begin with a local version of Proposition 5.8.

6.3 The Uniqueness, Mean-Value, and Maximum-Modulus Theorems

83

6.7 Proposition If f is analytic at α, so is ⎧ ⎨ f (z) − f (α) z−α g(z) = ⎩ f (α)

z = α z = α.

Proof By Theorem 6.6, in some neighborhood of α, f (z) = f (α) + f (α)(z − α) +

f (α) (z − α)2 + · · · . 2!

Thus g has the power series representation g(z) = f (α) +

f (3) (α) f (α) (z − α) + (z − α)2 + · · · 2! 3!

in the same neighborhood, and by 2.9, g is analytic at α.

6.8 Theorem If f is analytic at z, then f is infinitely differentiable at z. Proof We need only recall that, by definition, f is analytic at a point z if it is analytic in an open set containing z. By 6.6, then, in some disc containing z, f may be expressed as a power series. This completes the proof, since power series are infinitely differentiable (Corollary 2.10). 6.9 Uniqueness Theorem Suppose that f is analytic in a region D and that f (z n ) = 0 where {z n } is a sequence of distinct points and z n → z 0 ∈ D. Then f ≡ 0 in D. Proof Since f has a power series representation around z 0 , by the Uniqueness Theorem for Power Series, f = 0 throughout some disc containing z 0 . To show that f ≡ 0 in the whole domain D, we split D into two sets: A = {z ∈ D: z is a limit of zeroes of f }, B = {z ∈ D: z ∈ A}. By definition, A ∩ B = ∅. A is open by the Uniqueness Theorem for power series: if z is a limit of zeroes of f, f ≡ 0 in an entire disc around z and that disc is

84


contained in A. B is open since for each z ∈ B, there must be some δ > 0 such that f (ω) = 0 for 0 < |z − ω| < δ. The disc D(z; δ) would then be contained in B. By the connectedness of D, then, either A or B must be empty. But, by hypothesis, z 0 ∈ A. Thus B is empty and every z ∈ D is a limit of zeroes of f . By the continuity of f , then, f ≡ 0 in D. 6.10 Corollary If two functions f and g, analytic in a region D, agree at a set of points with an accumulation point in D, then f ≡ g through D. Proof Consider f − g.

Note that a non-trivial analytic function may have infinitely many zeroes. For example, sin z, which is entire, is equal to 0 at all the points z = nπ, n = 0, ±1, ±2, . . . . In fact, sin(1/z) = 0 on the set 1 : n = ±1, ±2, . . . nπ which has an accumulation point at 0! Because this limit point is not in the domain of analyticity of sin(1/z), however, sin(1/z) does not satisfy the hypothesis of Theorem 6.9. 6.11 Theorem If f is entire and if f (z) → ∞ as z → ∞, then f is a polynomial. Proof By hypothesis, there is some M > 0 such that |z| > M implies that | f (z)| > 1. We conclude that f has at most a finite number of zeroes α1 , α2 , . . . , α N . Otherwise, the set of zeroes would have an accumulation point in D(0; M), and by the Uniqueness Theorem f would be identically zero, contradicting the original hypothesis. If we divide out the zeroes of f , g(z) =

f (z) (z − α1 )(z − α2 ) · · · (z − α N )

is likewise entire (Corollary 5.9), and never equal to zero; hence h(z) =

1 = (z − α1 )(z − α2 ) · · · (z − α N )/ f (z) g(z)

is also entire. Since f → ∞ as z → ∞, |h(z)| ≤ A + |z| N ; therefore, by Theorem 5.11, h is a polynomial. But h = 1/g = 0, hence according to the


85

Fundamental Theorem of Algebra, h is a constant k. Thus f (z) =

1 (z − α1 )(z − α2 ) · · · (z − α N ). k

The Uniqueness Theorem is often used to demonstrate the validity in the complex plane of functional equations known to be true on the real line. For example, to prove the identity e z1 +z2 = e z1 e z2 (2) we first take z 2 to be a fixed real number. Then e z1 +z2 and e z1 · e z2 represent two entire functions of z 1 which agree at all real points and hence by the Uniqueness Theorem, they agree for all complex z 1 as well. Finally, for any fixed z 1 , we consider the two sides of (2) as analytic functions in z 2 which agree for real z 2 , and again applying the Uniqueness Theorem, we conclude that they agree for all complex z 2 as well. Hence (2) is valid for all complex z 1 and z 2 . Similarly, equations such as tan2 z = sec2 z − 1, which are known to be true for real z, are valid throughout their domains of analyticity. In general, if there is an “analytic” relationship among analytic functions: that is, a functional equation of the form F( f, g, h, . . .) = 0 which is satisfied by the analytic function F( f, g, h, . . .) on a set with an accumulation point in its region of analyticity, then the equation holds throughout the region. We now examine the local behavior of analytic functions. 6.12 Mean Value Theorem If f is analytic in D and α ∈ D, then f (α) is equal to the mean value of f taken around the boundary of any disc centered at α and contained in D. That is, f (α) =

1 2π

2π

f (α + r eiθ )dθ

0

when D(α; r ) ⊂ D. Proof This is a reformulation of the Cauchy Integral Formula (6.4) with a = α. That is, 1 f (z) dz, f (α) = 2πi Cr z − α

86


and introducing the parameterization z = α + r eiθ , we see that f (α) =

1 2π

2π

f (α + r eiθ )dθ.

0

In analogy with the real case, we will call a point z a relative maximum of f if | f (z)| ≥ | f (ω)| for all complex ω in some neighborhood of z. A relative minimum is defined similarly. 6.13 Maximum-Modulus Theorem A non-constant analytic function in a region D does not have any interior maximum points: For each z ∈ D and δ > 0, there exists some ω ∈ D(z; δ) ∩ D, such that | f (ω)| > | f (z)|. Proof The fact that | f (ω)| ≥ | f (z)| for some ω near z follows immediately from the Mean-Value Theorem. Since for r > 0 such that D(z; r ) ⊂ D we have f (z) =

1 2π

2π

f (z + r eiθ )dθ,

0

it follows that | f (z)| ≤

1 2π

2π 0

| f (z + r eiθ )|dθ ≤ max | f (z + r eiθ )|. θ

(3)

Similarly, we may deduce that | f (ω) | > | f (z)| for some ω ∈ D(z; r ). For, to obtain equality in (3), | f | would have to be constant throughout the circle C(z; r ) and since this holds for all sufficiently small r > 0, | f | would be constant throughout a disc. But then by Theorem 3.7, f would be constant in that disc, and by the Uniqueness Theorem, f would be constant throughout D. Ironically, the Maximum-Modulus Theorem actually asserts that an analytic function has no relative maximum. It is sometimes given a more positive flavor as follows. Suppose a function f is analytic in a bounded region D and continuous on ¯ (We will, henceforth, use the expression “ f is C-analytic in D” to denote D. ¯ the continuous function this hypothesis.) Somewhere in the compact domain D, | f | must assume its maximum value. The Maximum-Modulus Theorem may then be invoked to assert that this maximum is always assumed on the boundary of the domain.


87

6.14 Minimum Modulus Theorem If f is a non-constant analytic function in a region D, then no point z ∈ D can be a relative minimum of f unless f (z) = 0. Proof Suppose that f (z) = 0 and consider g = 1/ f . If z were a minimum point for f , it would be a maximum point for g. Hence g would be constant in D, contrary to our hypothesis on f . Remark We can also prove the Maximum-Modulus Theorem by analyzing the local power series representation for an analytic function. That is, for any point α, consider the power series f (z) = C0 + C1 (z − α) + C2 (z − α)2 + · · · , which is convergent in some disc around α. To find z near α and such that | f (z)| > | f (α)|, we first assume C1 = 0 and set z = α + δeiθ , with δ > 0 “small”, and θ chosen so that C0 and C1 δeiθ have the same argument. Then | f (α)| = |C0 | | f (z)| ≥ |C0 + C1 (z − α)| − |C2 (z − α)2 + C3 (z − α)3 + · · · | ≥ |C0 | + |C1 δ| − δ 2 |C2 + C3 (z − α) + · · · |. Since the last expression represents a convergent series, 1 | f (z)| ≥ |C0 | + |C1 δ| − Aδ 2 ≥ |C0 | + |C1 δ| > | f (α)| 2 as long as δ < |C1 |/2 A. Hence α cannot be a maximum point. Note that if C1 = 0, the same argument can be applied by focusing on the first non-zero coefficient Ck . This technique of studying the local behavior of an analytic function by considering the first terms of its power series expansion can be used to derive the following result. Recall that in calculus, relative maximum points were found among the critical points (those points at which f = 0) of a differentiable function f . The proposition below shows a somewhat surprising contrast in the behavior of an analytic function at a point where it assumes its maximum modulus. 6.15 Theorem Suppose f is nonconstant and analytic on the closed disc D, and assumes its maximum modulus at the boundary point z 0 . Then f (z 0 ) = 0.

88


Proof (G. Pólya and G. Szeg˝o) Assume that f (z 0 ) = 0. For any complex number ξ of sufficiently small modulus we have f (k) (z 0 ) k f (z 0 + ξ ) = f (z 0 ) + ξ + ··· , k! where k is the least integer with f (k) (z 0 ) = 0 and the omitted terms are all of higher order in ξ than ξ k . Multiplying the above expression by its conjugate shows | f (z 0 + ξ )|2 = f (z 0 + ξ ) f (z 0 + ξ ) = | f (z 0 )|2 +

2 Re f (z 0 ) f (k) (z 0 )ξ k + · · · . k!

Since | f (z 0 )| = maxz∈D | f (z)|, f (z 0 ) = 0. Write f (z 0 ) f (k) (z 0 ) = Aeiα with A > 0, and let eiθ = ξ/ |ξ |. Then | f (z 0 + ξ )|2 = | f (z 0 )|2 +

2A k |ξ | cos(kθ + α) + · · · , k!

and, for ξ of sufficiently small modulus, | f (z 0 + ξ )| − | f (z 0 )| has the same sign as cos(kθ + α). It follows that | f (z)| > | f (z 0 )| if z is in any of the k wedges of the form −π + 4π j − 2α π + 4π j − 2α , and rθ ∈ (0, εθ ) z 0 + rθ eiθ : θ ∈ 2k 2k

(4)

for some positive εθ and j = 0, 1, . . . , k − 1 (and | f (z)| < | f (z 0 )| if z is in any of the alternate wedges). Since f (z 0 ) = 0, k ≥ 2. To complete the proof, note that at least one of the k wedges described in (4) must intersect D. Hence | f (z 0 )| cannot be the maximum value of | f | on D. Remarks 1. While the theorem asserts that | f | cannot achieve an absolute maximum value at a critical point, it is equally true that | f | cannot have a minimum value other than zero at a critical point. This is obvious from the parenthetical remark after (4), above. It can also be proven by considering 1/ f (which is analytic on an open set containing D if f is nonvanishing on D). 2. Theorem 6.15 is easily generalized to a wide range of compact sets K , including those which do not have smooth boundaries. The key is that, along with each boundary point z 0 , K must also contain a wedge (or “cone”) of the form z 0 + r eiθ : θ ∈ [α, β] , r ∈ (0, ε)


89

with ε > 0 and β − α > π/2. This is sufficient since each of the wedges in (4) has a maximum vertex angle of π/2. Thus, the theorem would be equally valid for a polygon all of whose vertex angles were obtuse. Without this “cone condition”, however, the theorem is no longer valid. For example, in the unit square {z : Re z, Im z ∈ [0, 1]}, z 2 + i has both an absolute minimum and a critical point at 0, and 1/(z 2 + i ) has both an absolute maximum and a critical point at 0. 3. The ideas in the proof of Theorem 6.15 can be applied to show that the set of interior critical points of an analytic function (except for those which are also zeroes) is identical with the set of its “saddle points”. The details are given below. 6.16 Definition z 0 is a saddle point of an analytic function f if it is a saddle point of the real-valued function g = | f |; that is, if g is differentiable at z 0 ,with g x (z 0 ) = g y (z 0 ) = 0, but z 0 is neither a local maximum nor a local minimum of g. 6.17 Theorem z 0 is a saddle point of an analytic function f if and only if f (z 0 ) = 0 and f (z 0 ) = 0. Proof Let f = u + i v, where u and v are real, and let g = | f |. First, suppose that z 0 is a saddle point of f . Then g = | f | is differentiable at z 0 , and obviously g (z 0 ) = 0. Note that uu y + vv y (uu x + vv x ) , gy = . (5) gx = g g Since g x (z 0 ) = g y (z 0 ) = 0, u(z 0 )u x (z 0 ) + v(z 0 )v x (z 0 ) = 0, u(z 0 )u y (z 0 ) + v(z 0 )v y (z 0 ) = 0. u(z 0 ) and v(z 0 ) are not both 0, so the above equations imply that u x (z 0 ) v x (z 0 ) = 0. det u y (z 0 ) v y (z 0 ) From the Cauchy-Riemann equations, it follows that u 2x (z 0 )+v x2 (z 0 ) = 0, and hence that f (z 0 ) = 0. Conversely, if f (z 0 ) = 0, then u x (z 0 ) and v x (z 0 ) are both zero, and by the Cauchy-Riemann equations, the same is true for u y (z 0 ) and v y (z 0 ). It follows from (5) that g is differentiable with gx (z 0 ) = g y (z 0 ) = 0. However, as in the

90


proof of Theorem 6.15, the facts that f (z 0 ) = 0 and f (z 0 ) = 0 guarantee that z 0 is not an extremal point of g. Note: Of course, if f (z 0 ) = 0, | f | has an absolute minimum at z 0 . If, in addition, f (z 0 ) = 0, then it follows from the power series expansion of f about z 0 that, for z sufficiently close to z 0 and for some positive constant M, || f (z)| − | f (z 0 )|| ≤ | f (z) − f (z 0 )| ≤ M |z − z 0 |2 , showing that g = | f | is differentiable at z 0 with g x = g y = 0 there. If f (z 0 ) = 0 but f (z 0 ) = 0, it can be shown that | f | is not differentiable at z 0 . (See Bak-DingNewman) These observations can be illustrated by f (z) = (z − 1) (z − 4)2 , which has a simple zero at z = 1, a critical point but not a zero at z = 2, and a critical point at the double zero z = 4. The graph of | f | is shown in Figure 1. Note that | f | has a saddle point at z = 2 and is not differentiable at z = 1.

25 20 15 10 5 0 0

1 0 1

2

3

4

5 —1

Exercises 1. Find a power series expansion for 1/z around z = 1 + i. 1 by first using partial 2.* Find a power series, centered at the origin, for the function f (z) = 1−z−2z 2 fractions to express f (z) as a sum of two simple rational functions. 3. Using 1/(1 − z) = 1 + z + z 2 + · · · for |z| < 1, find closed forms for the sums nz n the2 identity n and n z .

4. Show that if f is analytic in |z| ≤ 1, there must be some positive integer n such that f (1/n) = 1/(n + 1). 5. Prove that sin(z 1 + z 2 ) = sin z 1 cos z 2 + cos z 1 sin z 2 . 6. Suppose an analytic function f agrees with tan x, 0 ≤ x ≤ 1. Show that f (z) = i has no solution. Could f be entire?

Exercises

91

7. Suppose that f is entire and that | f (z)| ≥ |z| N for sufficiently large z. Show that f must be a polynomial of degree at least N . 8. Suppose f is C-analytic √ in |z| ≤ 1, f 2 for |z| = 1, Im z ≥ 0 and f 3 for |z| = 1, Im z ≤ 0. Show then that | f (0)| ≤ 6. [Hint: Consider f (z) · f (−z).] 9. Show directly that the maximum and minimum moduli of ez are always assumed on the boundary of a compact domain. 10. Find the maximum and minimum moduli of z 2 − z in the disc: |z| ≤ 1. 11.* (A proof, due to Landau, of the maximum modulus theorem) Suppose f is analytic inside and on a circle C with | f (z)| ≤ M on C, and suppose z 0 is a point inside C. Use Cauchy’s integral formula to show that | f (z 0 )|n ≤ K M n , where K is independent of n, and deduce that | f (z 0 )| ≤ M. 12. Suppose f and g are both analytic in a compact domain D. Show that | f (z)| + |g(z)| takes its maximum on the boundary. [Hint: Consider f (z)eiα + g(z)eiβ for appropriate α and β.] 13. Show that the Fundamental Theorem of Algebra may be derived as a consequence of the MinimumModulus Theorem. 14. Suppose Pn (z) = a0 + a1 z + · · · + an z n is bounded by 1 for |z| ≤ 1. Show that |P(z)| ≤ |z|n for all z 1. [Hint: Use Exercise 6 of Chapter 5 to show |an | ≤ 1 and then consider P(z)/z n in the annulus: 1 ≤ |z| ≤ R for “large” R.] 15.* Let f (z) = (z − 1)(z − 4)2 . Find the lines (through z = 2) on which | f (z)| has a relative maximum, and the ones on which | f (z)| has a relative minimum, at z = 2. (See the figure at the end of the chapter.) 2 and identify the lines on which it is a relative maximum or 16.* Find the saddle point of f (z) = (z+1) z a relative minimum of | f |.

17.* a. Find the saddle points z 1 , z 2 of f (z) =

(z − 1)2 (z + 1) z2

b. Show that, for i = 1, 2 | f (z i )| = Max| f (z)| on the circle |z| = |z i |. c. Find lines through z i on which | f | has a relative maximum or a relative minimum at z i .

Chapter 7

Further Properties of Analytic Functions

7.1 The Open Mapping Theorem; Schwarz’ Lemma The Uniqueness Theorem (6.9) states that a non-constant analytic function in a region cannot be constant on any open set. Similarly, according to Proposition 3.7, | f | cannot be constant. Thus a non-constant analytic function cannot map an open set into a point or a circular arc. By applying the Maximum-Modulus Theorem, we can derive the following sharper result on the mapping properties of an analytic function. 7.1 Open Mapping Theorem The image of an open set under a nonconstant analytic mapping is an open set. Proof (due to Carathéodory). We will show that if f is non-constant and analytic at α, the image under f of some (small) disc containing α will contain a disc about f (α). Without loss of generality, assume f (α) = 0. (Otherwise, consider f (z) − f (α).) By the Uniqueness Theorem, there is a circle C around α such that f (z) = 0 for z ∈ C. Let 2 = minz∈C | f (z)|. It will follow that the image of the disc bounded by C contains the disc D(0; ). For assume that ω ∈ D(0; ) and consider f (z) − ω. For z ∈ C | f (z) − ω| ≥ | f (z)| − |ω| ≥ , while at α

| f (α) − ω| = | − ω| < .

93

94

7 Further Properties of Analytic Functions f(C) C

α

f

ω ε 0

Hence | f (z) − ω| assumes its minimum somewhere inside C, and by the Minimum Modulus Theorem, f (z) − ω must equal zero somewhere inside C. Thus ω is in the range of f . The Maximum-Modulus Theorem can also be used in conjunction with other given information about a function to obtain stronger estimates for the modulus of f in its domain of analyticity. The following example is typical. 7.2 Schwarz’ Lemma Suppose that f is analytic in the unit disc, that f 1 there and that f (0) = 0. Then i. | f (z)| ≤ |z| ii. | f (0)| ≤ 1 with equality in either of the above if and only if f (z) = eiθ z. Proof We apply the Maximum-Modulus Theorem to the analytic function ⎧ ⎨ f (z) 0 < |z| < 1 g(z) = z ⎩ f (0) z = 0. (See Proposition 6.7.) Since g 1/r on the circle of radius r , by letting r → 1 and applying the Maximum-Modulus Theorem, we find that |g(z)| ≤ 1 throughout the unit disc, proving (i) and (ii). Furthermore, if |g(z 0 )| = 1 for some z 0 such that |z 0 | < 1, then by the Maximum-Modulus Theorem, g would be a constant (of modulus 1), and f (z) = eiθ z.

7.1 The Open Mapping Theorem; Schwarz’ Lemma

95

A class of functions analytic in the unit disc and bounded there by 1 is given by the set of bilinear transformations Bα (z) = where |α| < 1. Note that

z−α 1 − αz ¯

1 > 1, α¯

so that Bα is analytic throughout |z| ≤ 1. On |z| = 1 |z|2 − α z¯ − αz ¯ + |α|2 z¯ − α¯ z−α 2 = =1 |Bα | = 1 − αz ¯ 1 − α z¯ 1 − α z¯ − αz ¯ + |α|2 |z|2 so that |Bα | ≡ 1 on the boundary. For this reason, the functions Bα can be used, in variations of Schwarz’ Lemma, to solve various extremal problems for analytic functions. E XAMPLE 1 Suppose that f is analytic and bounded by 1 in the unit disc and that f ( 12 ) = 0. We wish to estimate | f ( 34 )|. Since f ( 12 ) = 0, ⎧ z− 12 ⎪ ⎨ f (z) z = 12 1 g(z) = 1− 2 z ⎪ ⎩3 f 1 z = 12 4 2 is likewise analytic in |z| < 1. Letting |z| → 1, we find that |g| ≤ 1; so that z−1 2 | f (z)| ≤ 1 − 1z 2 throughout the disc. In particular, f 3 ≤ 2. 4 5 Note that the maximum value, 25 , is achieved by B 1 (z) = 2

z− 1−

1 2 . 1 2z

♦

E XAMPLE 2 Next we show that among all functions f which are analytic and bounded by 1 in the unit disc, max | f ( 13 )| is assumed when f ( 13 ) = 0.

96

7 Further Properties of Analytic Functions

Suppose f ( 13 ) = 0 and consider g(z) = Again, since

f (z) − f ( 13 ) 1 − f ( 13 ) f (z)

.

ω − f ( 13 ) =1 1 − f ( 13 )ω

when |ω| = 1, while | f | < 1 in |z| < 1, the Maximum-Modulus Theorem assures us that g, like f , is bounded by 1. A direct calculations shows that 2 ! 1 1 1 g = f 1 − f 3 3 3 so that

1 1 g >f . 3 3

We note that max | f ( 13 )| is assumed by the function B1/3 (z). [See Exercises 10 and 11.] ♦ Example 2 has an interesting physical interpretation. Given the constraint on f that it must map the unit disc into the unit disc, the way to maximize | f ( 13 )| is by a. mapping 13 into 0 and b. mapping the boundary of the unit disc onto itself. It is as though by thus allowing the maximum room for expansion around f ( 13 ), we obtain max | f ( 13 )|. We will see a similar phenomenon when we study the Riemann Mapping Theorem. Returning once again to entire functions, the Maximum-Modulus Theorem may be used to derive further extensions of Liouville’s Theorem. 7.3 Proposition If f is an entire function satisfying | f (z)| ≤ 1/|Im z| for all z, then f ≡ 0. Proof By hypothesis f 1 throughout |Im z| > 1 but f could be unbounded near the real axis. To estimate | f | on the circle |z| = R, we introduce the auxiliary function g(z) = (z 2 − R 2 ) f (z).

7.1 The Open Mapping Theorem; Schwarz’ Lemma

97

For any z such that |z| = R and Re z ≥ 0 |(z − R) f (z)| ≤ |z − R|/|Im z| = sec θ for some θ , 0 ≤ θ ≤ π/4 (see the following diagram), so that √ |(z − R) f (z)| ≤ 2.

z

θ

0

R

Similarly, if |z| = R and Re z ≤ 0, then |(z + R) f (z)| ≤

√

2.

Thus |g(z)| = |z + R||z − R|| f (z)| ≤ 3R for all z with |z| = R. By the Maximum-Modulus Theorem, the same upper bound holds throughout |z| < R. Hence |g(z)| = |z 2 − R 2 || f (z)| ≤ 3R and | f (z)| ≤

3R |z 2 − R 2 |

as long as R z. Letting R → ∞, we see that f (z) = 0. Since this holds for all z, the theorem is proven.

98


7.2 The Converse of Cauchy’s Theorem: Morera’s Theorem; The Schwarz Reflection Principle and Analytic Arcs The key result in our study of analytic functions so far has been the Rectangle Theorem (6.1). Thus, it may not come as a surprise that the property described there is almost equivalent to analyticity. 7.4 Morera’s Theorem Let f be a continuous function on an open set D. If f (z)dz = 0

whenever is the boundary of a closed rectangle in D, then f is analytic on D. Since line integrals are unaffected by the value of the integrand at a single point, the continuity of f is a necessary hypothesis. Note also that in the proof, we actually require only that f = 0 for rectangles whose sides are parallel to the horizontal and vertical axes. Proof In a small disc about any point z 0 ∈ D, we can define a primitive z f (ζ )dζ F(z) = z0

where the path of integration is the horizontal followed by the vertical segments from z 0 to z. If we then consider a difference quotient of F and apply the fact that f = 0 around any rectangle, we may conclude (as in Theorems 4.15 and 6.2) that 1 z+h F(z + h) − F(z) = f (ζ )dζ → f (z) h h z as h → 0. (Here we are using the continuity of f .) Hence F is analytic in a neighborhood of z 0 . Since analytic functions are infinitely differentiable and F (z) = f (z), f is analytic at z 0 . Finally, since z 0 was arbitrary, f is analytic in D. Morera’s Theorem is often used to establish the analyticity of functions given in integral form. For example, consider ∞ zt e dt. f (z) = t +1 0 If Re z = x < 0,

0

∞

|e zt | dt < t +1

∞ 0

e xt dt = −

1 x

7.2 The Converse of Cauchy’s Theorem: Morera’s Theorem

99

so that the integral is absolutely convergent and | f (z)| ≤ 1/|x|. To show that f is analytic in the left half-plane D : Re z < 0, we may consider ∞ zt e dt dz, f (z)dz = t +1 0 where is the boundary of some closed rectangle in D. Since ∞ zt |e | dt dz t +1 0 converges, we can interchange the order of integration; hence ∞ ∞ e zt dz dt = f = 0 dt = 0 0 0 t +1 by the analyticity of e zt /(t + 1) as a function of z. By Morera’s Theorem, then, f is analytic in D. 7.5 Definition Suppose { f n } and f are defined in D. We will say f n converges to f uniformly on compacta if f n → f uniformly on every compact subset K ⊂ D. The following theorem asserts that analyticity is preserved under uniform limits, in marked contrast to the property of differentiability on the real line. There, the uniform limit of differentiable functions may be nowhere differentiable. 7.6 Theorem Suppose { f n } represents a sequence of functions, analytic in an open domain D and such that f n → f uniformly on compacta. Then f is analytic in D. Proof In some compact neighborhood K of each point z 0 , f is the uniform limit of continuous functions; hence f is continuous in D. Furthermore, for every rectangle ⊂K f = lim f n = lim f n = 0,

n

since fn → f uniformly on . Hence, by Morera’s theorem, f is analytic in D. 7.7 Theorem Suppose f is continuous in an open set D and analytic there except possibly at the points of a line segment L. Then f is analytic throughout D.

100


Proof Without loss of generality, we may assume the exceptional points lie on the real axis. Otherwise, we could begin by considering g(z) = f (Az + B) where Az + B maps the real axis onto the line containing L. (See Exercise 15.) Of course, the analyticity of f on D is equivalent to the analyticity of g on the corresponding region. Moreover, since analyticity is a local property, we may assume D is a disc. To show f = 0 for every closed rectangle in D with boundary (and with sides parallel to the real and imaginary axes), we consider three cases. i. L doesn’t meet the rectangle bounded by . Here f = 0 by the analyticity of f throughout the interior of (Theorem 6.1). ii. One side of coincides with L. In this case, we let be the rectangle composed of the sides of with the bottom (or top) side shifted up (or down) by in the positive

D

Γ

Γε 0

a

(ii)

b

Γ1

(iii) 0 Γ2

(or negative) y-direction. Then

f = lim

→0

f,


since

b

a

f (x)d x

a

by the continuity of f . Hence

iii. If surrounds L, we write

b

f (x + i )d x →

101

f = 0,

f =

1

f +

2

f

where 1 and 2 are as in (ii). Again we conclude f = 0.

Finally, By Morera’s Theorem, f is analytic in D.

A wide range of results, all of which are known as the Schwarz Reflection Principle, are typified by the following theorem.

7.8 Schwarz Reflection Principle Suppose f is C-analytic in a region D that is contained in either the upper or lower half plane and whose boundary contains a segment L on the real axis, and suppose f is real for real z. Then we can define an analytic “extension” g of f to the region D ∪ L ∪ D ∗ that is symmetric with respect to the real axis by setting f (z) z ∈ D ∪ L g(z) = f (¯z ) z ∈ D ∗ where D ∗ = {z : z¯ ∈ D}.

D L D*

102


Proof At points in D, g = f and hence g is analytic there. If z ∈ D ∗ and h is small enough so that z + h ∈ D ∗ ¯ − f (¯z ) f (¯z + h) ¯ − f (¯z ) g(z + h) − g(z) f (¯z + h) = = h h h¯ which approaches f (¯z ) as h approaches 0. Hence g is analytic in D ∗ . Since f is continuous on the real axis, so is g and we can apply Theorem 7.7 to conclude that g is analytic throughout the region D ∪ L ∪ D ∗ . By invoking the Uniqueness Theorem, we obtain the following immediate corollary: 7.9 Corollary If f is analytic in a region symmetric with respect to the real axis and if f is real for real z, then f (z) = f (¯z ). The Schwarz reflection principle can be applied in more general situations. The key is to extend the concept of reflection across other curves. 7.10 Definition A curve γ : [a, b] → C will be called a regular analytic arc if γ is an analytic, one-to-one function on [a, b] with γ = 0. Note that, by the definition of analyticity, γ is the restriction to [a, b] of a function γ (z) which is analytic in an open set S containing [a, b]. Moreover, if all points of S are sufficiently close to [a, b], γ = 0 and γ will remain one-to-one throughout S. (Otherwise, the original curve would fail to be one-to-one or γ would be zero at some point of [a, b].) So assume that γ (z) is analytic and one-to-one in such an open set S which is also symmetric with respect to the interval [a, b]. Then we can define the reflection w∗ of a point w in γ (S), across the curve γ , as γ (γ −1 (w)). That is, if w = γ (z), w∗ = γ (z). It follows immediately that (w∗ )∗ = w, that points on the original curve are reflected into themselves, and that points not on the curve γ are reflected onto other points not on γ . In fact, the arc formed by taking the image under γ of the vertical line from any nonreal z to its conjugate z must intersect the original curve γ (i.e. γ (t), a ≤ t ≤ b) orthogonally, by the conformality of γ . Hence w and w∗ are on opposite sides of γ . ∗ For example, if γ (t) = i t, −∞ < t < ∞, and w = u + i v, then w = γ v − i u = −u + i v = −w, ¯ which is the reflection of w across the imaginary axis. Similarly, suppose γ is an arc of the circle γ (t) = Reit . Then γ (z) = Reiz = Re−y ei x . If w = γ (z) = r eiθ , Re−y = r and x = θ, so that w∗ = γ (x − i y) =


103

R2 R 2 iθ e = . Note that w∗ is on the same ray as w, and |ww∗ | = R 2 , so that w and r∗ w¯ w are on opposite sides of the circle of radius R. Suppose then that f is analytic in a region S and continuous to the boundary, which includes the regular analytic curve γ , and assume that f (γ ) ⊂ λ, another regular analytic curve. Let z ∗ denote the reflection of z across γ , and let wˆ denote the reflection of w across λ. Then f can be extended to S ∗ by defining f (z) at a point z ∈ S ∗ as ( f (z ∗ ))ˆ . This defines an analytic extension of f to S ∗ since it is equal to the composition: λ ◦ λ−1 ◦ f ◦ γ ◦ γ −1 . As in our proof of the original form of the Schwarz reflection principle, the analyticity of f follows from the fact that h(z) is analytic at z (and has a derivative equal to h (z) ) whenever h is analytic at z.

S

γ

f

f(S)

λ

Example 1: Suppose f is analytic in the unit disc and continuous to the boundary, which it maps into itself. Then f can be extended by defining f (z) = 1/ f (1/z) at points z outside the unit circle. Note that the extended function is analytic everywhere except at the reflections of the zeroes of f inside the unit circle, which the extended function would map into ∞. Thus if we were looking for a bilinear function f mapping the unit circle into itself, with f (α) = 0, it would follow that f (1/α) = ∞, so that we might consider f (z) = (z − α)/ (z − 1/α) . However, in its current form f does not map the unit circle into itself. In particular, | f (1)| = |α|, so we must multiply our function by a constant of magnitude 1/|α|, which leads us to consider functions of the form f (z) = (z − α)/ (1 − αz). As we saw in the last section, these bilinear functions do, in fact, map the unit circle into itself. ♦ Example 2: Suppose f is an analytic map of a rectangle R onto another rectangle S, which maps each side of R onto a side of S. Then f can be extended analytically across the sides of R, mapping rectangles adjacent to R onto rectangles adjacent to S. Continuing in this manner, f can be extended to an entire function! It is easily seen, moreover, that the extended entire function has “linear growth”; i.e. | f (z)| ≤ A|z| + B, for some positive constants A and B. Hence, according to the Extended Liouville Theorem, f must be a linear polynomial. ♦

104


Exercises 1. Show that if f is analytic and non-constant on a compact domain, Re f and Im f assume their maxima and minima on the boundary. 2. Prove that the image of a region under a non-constant analytic function is also a region. 3. a. Suppose f is nonconstant and analytic on S and f (S) = T . Show that if f (z) is a boundary point of T , z is a boundary point of S. b. Let f (z) = z 2 on the set S which is the union of the semi-discs S1 = {z : |z| ≤ 2; Re z ≤ 0} and S2 = {z : |z| ≤ 1; Re z ≥ 0}. Show that there are points z on the boundary of S for which f (z) is an interior point of f (S). 4. Suppose f is C-analytic in D(0; 1) and maps the unit circle into itself. Show then that f maps the entire disc onto itself. [Hint: Use the Maximum-Modulus Theorem to show that f maps D(0; 1) into itself. Then apply the previous exercise to conclude that the mapping is onto.] 5. Suppose f is entire and | f | = 1 on |z| = 1. Prove f (z) = C z n . [Hint: First use the maximum and minimum modulus theorem to show f (z) = C

n " z − αi .] 1 − α¯ i z

i=1

6.* Show that for any given rational function f (z), with poles in the unit disc, it is possible to find another rational function g(z), with no poles in the unit disc, and such that | f (z)| = |g(z)| if |z| = 1. 7.* a. Suppose |α| < R. Show that

R(z − α) R 2 − αz

is analytic for |z| ≤ R, and maps the circle |z| = R into the unit circle. b. Suppose |αk | < R for k = 1, 2, ...n. Prove that (unless |αk | = 0 for all k) n |z − α1 | · |z − α2 | · · · |z − αn | assumes a maximum value greater than R, and a minimum value less than R, at some points z n " on |z| = R. [Hint: Apply the maximum and minimum modulus theorems to (R 2 − αk z).] k=1

8. Suppose that f is analytic in the annulus: 1 ≤ |z| ≤ 2, that | f | ≤ 1 for |z| = 1 and that | f | ≤ 4 for |z| = 2. Prove | f (z)| ≤ |z|2 throughout the annulus. 9. Given f analytic in |z| < 2, bounded there by 10, and such that f (1) = 0. Find the best possible upper bound for | f ( 12 )|. 10. Suppose that f is analytic and bounded by 1 in the unit disc with f (α) = 0 for some α 1. Show that there exists a function g, analytic and bounded by 1 in the unit disc, with |g (α)| > | f (α)|. 11. Find max f | f (α)| where f ranges over the class of analytic functions bounded by 1 in the unit disc, and α is a fixed point of |z| < 1. [Hint: By the previous exercise, you may assume f (α) = 0.] Show that f (z) Bα (z) lim = Bα (α). f (α) = lim z→α z − α z→α z − α 12. Suppose f is entire and | f (z)| ≤ 1/|Re z|2 for all z. Show that f ≡ 0.

Exercises

105

13. Show that f (z) =

1 sin zt dt t 0

is an entire function. a. by applying Morera’s Theorem, b. by obtaining a power series expansion for f . 14. With f as in (13) show that

1

f (z) =

cos ztdt 0

a. by writing f (z) = =

1 z 0

0

cos zt dz dt

z 1

cos zt dt dz, etc., 0

0

b. by using the power series for f . 15. Show that g(z) = z 0 + eiθ z, θ = Arg(z 1 − z 0 ), maps the real axis onto the line L through z 0 and z 1 . 16. Suppose f is bounded and analytic in Im z ≥ 0 and real on the real axis. Prove that f is constant. 17. Given an entire function which is real on the real axis and imaginary on the imaginary axis, prove that it is an odd function: i.e., f (z) = − f (−z). 18.* Show that v + iu is the reflection of the point u + iv across the line u = v. 19. Suppose f is analytic in the semi-disc: |z| ≤ 1, Im z > 0 and real on the semi-circle |z| = 1, Im z > 0. Show that if we set ⎧ ⎪ |z| ≤ 1, Im z > 0 ⎨ f (z) g(z) = ⎪ |z| > 1, Im z > 0 ⎩f 1 z¯ then g is analytic in the upper half-plane. 20. Show that there is no non-constant analytic function in the unit disc which is real-valued on the unit circle. 21. Suppose f is analytic in the upper semi-disc: |z| ≤ 1, Im z > 0 and is continuous to the boundary. Explain why it is not possible that f (x) = |x| for all real values of x. 22.* Suppose an entire function maps two horizontal lines onto two other horizontal lines. Prove that its derivative is periodic. [Hint: Assume f = u + iv maps the lines y = y1 and y = y2 onto v = v 1 and v = v 2 with y2 − y1 = c and v 2 − v 1 = d. Show then that f (z + 2ci) = f (z) + 2di, for all z.] 23.* Prove that an entire function which maps a parallelogram onto another parallelogram, and maps each side of the original parallelogram onto a side of its image, must be a linear polynomial. [Hint: Use Exercise 22 to prove that f is constant.]

Chapter 8

Simply Connected Domains

8.1 The General Cauchy Closed Curve Theorem As we have seen, it can happen that a function f is analytic on a closed curve C and yet C f = 0. Perhaps the simplest such example was given by 1 dz = 2πi. |z|=1 z On the other hand, the Closed Curve Theorem—6.3—showed that if f is analytic throughout a disc, the integral around any closed curve is 0. We now seek to determine the most general type of domain in which the Closed Curve Theorem is valid. Note that the domain of analyticity of f (z) = 1/z is the punctured plane. We will see that it is precisely the existence of a “hole” at z = 0 which allowed the above counterexample. The property of a domain which assures that it has no “holes” is called simple connectedness. The formal definition is as follows.

8.1 Definition A region D is simply connected if its complement is “connected within to ∞.” That is, if for any z 0 ∈ D˜ and > 0, there is a continuous curve γ (t), 0 ≤ t < ∞ such that ˜ < for all t ≥ 0, (a) d(γ (t), D) (b) γ (0) = z 0 , (c) limt →∞ γ (t) = ∞. A curve γ, satisfying (b) and (c), is said to “connect z 0 to ∞.” (See Chapter 1.4.) E XAMPLE 1 The plane minus the real axis is not simply connected since it is not a region; that is, a simply connected domain must be connected. ♦ 107

108

8 Simply Connected Domains

E XAMPLE 2 The annulus A = {z : 1 < |z| < 3} is not simply connected.

1

3

To prove this, note that 0 ∈ A˜ and yet there is no γ which remains within = 12 of A˜ and connects 0 to ∞. If such a γ existed, by the continuity of |γ (t)|, there would ˜ = 1. have to be a point t1 such that |γ (t1 )| = 2, but then d(γ (t1 ), D) ♦ E XAMPLE 3 The unit disc minus the positive real axis is simply connected since for any z 0 in the complement γ : γ (t) = (t + 1)z 0 connects z 0 to ∞ and is contained in the complement.

♦

8.1 The General Cauchy Closed Curve Theorem

109

E XAMPLE 4 The infinite strip S = {z : − 1 < Im z < 1} is simply connected. Note that in this case, the complement S˜ is not connected. y

i

x

–i

♦ E XAMPLE 5 Any open convex set is simply connected. See Exercises 1 and 2.

♦

Definition 8.1 requires some explanation. It may seem somewhat simpler to say a region D is simply connected if every point in its complement can be connected, by a curve in the complement, to ∞. However, although this is the case in all the above examples, it is still somewhat too restrictive. For example, suppose the complement is the (connected) set 0<x ≤1 D˜ = x + i y: ∪ {i y : − 1 ≤ y < ∞}. y = sin 1x By Definition 8.1, D would then be simply connected although the points on the ˜ For a comparicurve y = sin(1/x) cannot be connected to ∞ by a curve in D. son of Definition 8.1 with other definitions of simple connectedness, see [Newman, pp. 164ff]. Also, see Appendix I. Before proving the general closed curve theorem, we first prove an analogue for simple closed polygonal paths. Recall that a polygonal path is a finite chain of horizontal and vertical line segments. 8.2 Definition Let be a polygonal path. We define the number of levels of as the number of different values y0 for which the line Im z = y0 contains a horizontal

110


segment of .

2 levels

3 levels

4 levels

8.3 Lemma Let be a simple closed polygonal path contained in a simply connected domain D. Suppose the top level of consists of the points y = y1 , x ∈ X 1 and the next level ≤y≤y1 is given by y = y2 , x ∈ X 2 . Then the set R = {z = x + i y: y2x∈X } is contained 1 in D. Proof Note that R is a finite union of disjoint closed rectangles. We will show that for any z 0 ∈ R and any curve γ connecting z 0 to ∞, γ ∩ = ∅. Then, since D˜ is closed ˜ = δ > 0, and γ would not remain within = δ/2 of D. and is compact, d(, D) Thus z 0 ∈ D. z0

L

R

γ

To show γ ∩ = ∅, we proceed by induction on the number of levels of . If has only two levels, it is the boundary of a single rectangle and the proof is straightforward (the details are given in Exercise 5). Otherwise we consider L = {x + i y : y = y2 , x ∈ X 1 \X 2 }. Note that z 0 is contained in one of the rectangles of R, so γ must intersect the boundary of R. Thus, if γ doesn’t meet R ∩ , it must meet L. Setting t0 = sup{t : γ (t) ∈ R}

8.1 The General Cauchy Closed Curve Theorem

111

we note that for small enough h > 0, γ (t0 + h) would be between the top two levels of a simple closed polygonal curve which is a connected component of ˜ ∪ L¯ = ( ∩ R) and has one less level than . But then, by induction γ (t) ∈ for some t > t0 + h. Finally, since γ (t) ∈ R for t > t0 and since L ⊂ R, γ (t) ∈ and the proof is complete. 8.4 Theorem Suppose f is analytic in a simply connected region D and is a simple closed polygonal path contained in D. Then f = 0. Proof of Theorem 8.4 The proof will again be by induction on the number of levels of . Define L, R and as in the lemma. We can write f = f + f

∂R

the integral over L being taken in opposite directions. Since ∂ R consists of the boundaries of rectangles and since f is analytic throughout these rectangles (by the lemma), ∂ R f = 0 by the Rectangle Theorem (6.1). Proceeding by induction on the number of levels of , we may assume f =0

since it has one less level than . Hence

f = 0 and the proof is complete.

8.5 Theorem If f is analytic in a simply connected region D, there exists a “primitive” F, analytic in D and such that F = f . Proof Choose z 0 ∈ D and define

F(z) =

z

f (ζ )dζ,

z0

where the path of integration is a polygonal path contained in D. By the previous theorem, F is well-defined for if we take 1 and 2 to be two such polygonal paths from z 0 to z, f − f = f 1

2

112


where is a closed polygonal curve. We leave it as an exercise to show that any closed polygonal curve can be decomposed into a finite number of simple closed polygonal curves and line segments traversed twice in opposite directions. Thus it follows from Lemma 8.3 that f = 0 and 1 f = 2 f . To show that F = f , we consider 1 F(z + h) − F(z) = h h

z+h

f (ζ )dζ

z

where now (by taking h small enough), we may take the simplest path of integration: horizontally and then vertically from z to z + h. It follows, then, as in Theorems 5.2 and 6.2, that F (z) = f (z). 8.6 General Closed Curve Theorem Suppose that f is analytic in a simply connected region D and that C is a smooth closed curve contained in D. Then f = 0. C

Proof

F (z)dz

f = C

C

where F is the primitive function guaranteed by Theorem 8.5, = F(z(b)) − F(z(a)) = 0 since the endpoints of the closed curve coincide.

It might be noted that while Theorem 8.6 is stated for simply connected regions, it has implications for other domains as well. For example, if f is analytic in the punctured plane z = 0 and C is a closed curve in the upper half-plane, C f = 0 since C may be viewed as a closed curve in the simply connected subset Im z > 0, where f is analytic. In general, if f is analytic in D and if C is contained in a simply connected subset of D, then C f = 0. E XAMPLE 1 Suppose C is the circle α + r eiθ , 0 ≤ θ ≤ 2π and |a − α| > r . Then dz =0 z C −a since 1/(z − a) is analytic in the simply connected disc: |z − α| < |a − α| which contains C. (Compare with Lemma 5.4.) ♦

8.2 The Analytic Function log z

113

Cauchy’s Theorem also allows us at times to switch an integral along one closed contour to another. E XAMPLE 2 Suppose f is analytic in the annulus: 1 ≤ |z| ≤ 4. Then f (z)dz = f (z)dz |z|=2

|z|=3

since, by adding the integrals along the real axis from 2 to 3 and from −2 to −3 in both directions, we can write f (z)dz − f (z)dz = f (z)dz + f (z)dz |z|=3

|z|=2

1

2

where 1 and 2 are closed curves contained in simply connected subsets of the annulus. (See below.)

Γ1

–3

–2

2

3

Γ2

♦

8.2 The Analytic Function log z 8.7 Definition We will say f is an analytic branch of log z in a domain D if (1) f is analytic in D, and (2) f is an inverse of the exponential function there; i.e., exp( f (z)) = z. Of course if f is an analytic branch of log z then so is g(z) = f (z) + 2πki for any fixed integer k.

114


Since eω = 0 for any ω, log 0 is not defined. However, for any z = Reiθ , R > 0, if we set f (z) = log z = u(z) + i v(z) condition (2) above becomes exp( f (z)) = eu(z) · eiv(z) = Reiθ which is possible if and only if (3) eu(z) = |z| = R and (4) v(z) = Arg z = θ + 2kπ. Hence a function f satisfying (2) can always be found by setting (5) f (z) = u(z) + i v(z) = log |z| + i Arg z. However, Arg z is not a well-defined function [see Chapter 1.2] and even if we adopt a particular convention for Arg z, it is not clear that the function defined in (5) is analytic (or even continuous) in D. However, if D is a simply connected domain not containing 0, we may define an analytic branch of log z there. (Recall that according to Theorem 3.5 if an analytic inverse of e z exists, its derivative must be 1/z. Thus we proceed as follows.) 8.8 Theorem Suppose that D is simply connected and that 0 ∈ D. Choose z 0 ∈ D, fix a value of log z 0 and set z dζ + log z 0 . (6) f (z) = z0 ζ Then f is an analyti c branch of log z in D. Proof f is well-defined since 1/ζ is an analytic function of ζ in D and hence the integral along any two paths from z 0 to z yield the same value (Theorem 8.5). Furthermore, f (z) = 1/z, so f is analytic in D. To show that exp( f (z)) = z, we consider g(z) = ze− f (z) . Since g (z) = e− f (z) − z f (z)e− f (z) = 0, g is constant and g(z) = g(z 0 ) = z 0 e− f (z0 ) = 1. Hence

e f (z) = z.

8.2 The Analytic Function log z

115

In an analogous manner, we can define an analytic branch of log f (z) in any simply connected domain where f is analytic and unequal to 0. We simply fix z 0 and a value of log f (z 0 ) and set z f (ζ ) log f (z) = dζ + log f (z 0 ). z 0 f (ζ ) In a typical situation, suppose D represents the whole plane minus the non-positive real axis: x ≤ 0. If we choose z 0 = 1 and log 1 = 0 in (6) the resulting function, z dζ , f (z) = 1 ζ is an analytic branch of log z with −π < Im(log z) = Arg z < π. (This latter inequality can be seen by integrating from 1 to |z| and from |z| to z.) Similarly, if D is the plane slit along the non-negative real axis and we choose that branch of log z for which log(−1) = πi , we will have defined an analytic branch of log z with 0 < Arg z < 2π. [See Exercise 8.] By proper application of the logarithm, we can also define analytic branches of √ 1/3 z, z , etc., in the appropriate domains. √ For example, z may be defined, in any domain where log z is defined, as √ (7) z = exp( 12 log z). Since

2 1 log z = exp(log z) = z, exp 2

√ this does define a “ z” and it is analytic where the logarithm is. Note that different √ branches of log z may yield different branches of z. Unlike log z, however, which has infinitely many different branches log z + 2πki √ for any integer k, there are only two different branches of z. This follows from the fact that the equation w2 = z has exactly two different solutions for any z = 0. It also follows from (7) since 1 1 exp log z = exp [ log z + 2πki ] 2 2 if k is even. The same technique may be used to define arbitrary powers of any nonzero complex number. For example, i i ≡ ei log i = {. . . e3π/2 , e−π/2 , e−5π/2 , . . .}.

116


Exercises 1. A set S is called star-like if there exists a point α ∈ S such that the line segment connecting α and z is contained in S for all z ∈ S. Show that a star-like region is simply connected. [Hint: Show that γ : γ (t) = tz + (1 − t)α, t ≥ 1 is contained in the complement for any z in the complement.] 2.* Prove that every convex region is simply connected. 3. Suppose a region S is simply connected and contains the circle C = {z : |z − α| = r}. Show then that S contains the entire disc D = {z : |z − α| ≤ r}. [Hint: Show that since S is open (by definition) and C is compact, S contains the annulus B = {z : r − δ ≤ |z − α| ≤ r + δ} for some δ > 0.] 4. Show that if

0<x ≤1 ∪ {iy : − 1 ≤ y < ∞}, s˜ = x + iy : 1 y = sin x

S is simply connected. 5.* Show that a polygonal line γ connecting z to ∞ intersects the boundary of every rectangle R containing z. [Hint: Consider t0 = sup{t : γ (t) ∈ R}.] 6. Define the “inside” of a simple closed polygonal path. Show that if such a path is contained in a simply connected domain, so is its “inside.” 7. Show that any closed polygonal path can be decomposed into a finite union of simple closed polygonal paths and line segments traversed twice in opposite directions. z dζ /ζ defines an analytic branch of log z in the plane slit along the non-negative 8. Show that πi + −1 axis with 0 < Im log z = Arg z < 2π . 9.* Define a function f analytic in the plane minus the non-positive real axis and such that f (x) = x x on the positive axis. Find f (i), f (−i). Show that f (¯z ) = f (z) for all z.

Chapter 9

Isolated Singularities of an Analytic Function

9.1 Classification of Isolated Singularities; Riemann’s Principle and the Casorati-Weierstrass Theorem Introduction While we have concentrated until now on the general properties of analytic functions, we now focus on the special behavior of an analytic function in the neighborhood of an “isolated singularity.” We will use the term deleted neighborhood of z 0 to denote a set of the form {z : 0 < |z − z 0 | < d}. 9.1 Definition f is said to have an isolated singularity at z 0 if f is analytic in a deleted neighborhood D of z 0 but is not analytic at z 0 . Note that, by Theorem 7.7, f must be discontinuous at an isolated singularity. E XAMPLES

sin z z = 2 0 z=2 has an isolated singularity at z = 2. ii. g(z) = 1/(z − 3) has an isolated singularity at z = 3. iii. exp(1/z) has an isolated singularity at z = 0. i. The function defined by f (z) =

♦

As we shall soon see, the above examples represent the different types of isolated singularities. These may be classified as follows. 9.2 Definition Suppose f has an isolated singularity at z 0 . i. If there exists a function g, analytic at z 0 and such that f (z) = g(z) for all z in some deleted neighborhood of z 0 , we say f has a removable singularity at z 0 (i.e., if the value of f is “corrected” at the point z 0 , it becomes analytic there). 117

118

9 Isolated Singularities of an Analytic Function

ii. If, for z = z 0 , f can be written in the form f (z) = A(z)/B(z) where A and B are analytic at z 0 , A(z 0 ) = 0, and B(z 0 ) = 0, we say f has a pole at z 0 . (If B has a zero of order k at z 0 , we say that f has a pole of order k.) iii. If f has neither a removable singularity nor a pole at z 0 , we say f has an essential singularity at z 0 . The following theorems show how the nature of the singularity possessed by a function may be determined by its behavior in a deleted neighborhood of the singularity. 9.3 Riemann’s Principle of Removable Singularities If f has an isolated singularity at z 0 and if limz→z0 (z − z 0 ) f (z) = 0, then the singularity is removable. Proof (z − z 0 ) f (z) z = z 0 h(z) = 0 z = z0.

Consider

By hypothesis, h is continuous at z 0 . Since h, like f , is analytic in a deleted neighborhood of z 0 , it follows that h is analytic at z 0 (Theorem 7.7). Since h(z 0 ) = 0, g(z) = h(z)/(z − z 0 ) is likewise analytic at z 0 and equals f for z = z 0 . 9.4 Corollary If f is bounded in a deleted neighborhood of an isolated singularity, the singularity is removable. 9.5 Theorem If f is analytic in a deleted neighborhood of z 0 and if there exists a positive integer k such that lim (z − z 0 )k f (z) = 0 but

z→z 0

lim (z − z 0 )k+1 f (z) = 0,

z→z 0

then f has a pole of order k at z 0 . Proof If we set

(z − z 0 )k+1 f (z) z = z 0 g(z) = 0 z = z0

then g is continuous and hence analytic at z 0 . Furthermore, since g(z 0 ) = 0, A(z) =

g(z) = (z − z 0 )k f (z) z − z0

9.1 Classification of Isolated Singularities; Riemann’s Principle

119

is likewise analytic at z 0 , and by hypothesis A(z 0 ) = 0. Since f (z) =

A(z) for z = z 0 (z − z 0 )k

the proof is complete.

Note that according to the previous two theorems, there is no analytic function which approaches ∞ like a fractional power of 1/(z − z 0 ) in the neighborhood of an isolated singularity z 0 . For √example, if f were analytic in a deleted neighborhood of 0 and satisfied | f (z)| ≤ 1/ |z|, then by 9.3, f would be bounded since the singularity would be removable. Similarly, given that | f (z)| ≤

1 , |z|5/2

we conclude that z 2 f (z) has a removable singularity at 0. Hence f has a pole of order at most 2 at the origin and, in fact, f (z) ≤ A/|z|2 . It also follows that in the neighborhood of an essential singularity, a function f must be not only unbounded but such that, for each integer N, (z − z 0 ) N f (z) → / 0 as z → z 0 . It does not follow, however, that f (z) → ∞ as z → z 0 . In fact, the following theorem shows that the set of values assumed by a function in the neighborhood of an essential singularity is “dense” in the whole complex plane. That is, the range of f intersects every disc in C. 9.6 Casorati-Weierstrass Theorem If f has an essential singularity at z 0 and if D is a deleted neighborhood of z 0 , then the range R = { f (z):z ∈ D} is dense in the complex plane. Proof Assume there exists some disc with center ω and radius δ which does not intersect R.

δ ω

120


Then | f (z) − ω| > δ and 1 1 f (z) − ω < δ throughout D. By Riemann’s Principle (9.3), it follows that 1/( f (z) − ω) has (at most) a removable singularity at z 0 . Hence 1 = g(z) f (z) − ω where g is analytic at z 0 . But then f (z) = ω +

1 g(z)

so that f has either a pole (if g(z 0 ) = 0) or a removable singularity (if g(z 0 ) = 0) at z 0 . There is, in fact, a much stronger form of the Casorati-Weierstrass Theorem– known as Picard’s Theorem–which asserts that an analytic function takes every value with at most a single exception in the neighborhood of an essential singularity.

9.2 Laurent Expansions In Chapter 6, we saw that functions analytic in a disc could be represented there by power series. A somewhat similar representation–by “two-sided power series” of k the form ∞ k=−∞ ak (z − z 0 ) –can be derived for functions analytic in an annulus R1 < |z − z 0 | < R2 . These two-sided power series, known as Laurent expansions, are valuable tools in the study of isolated singularities. 9.7 Definition ∞ ∞ We say ∞ k=−∞ μk = L if both k=0 μk and k=1 μ−k converge and if the sum of their sums is L. 9.8 Theorem k f (z) = ∞ −∞ ak z is convergent in the domain D = {z : R1 < |z| and |z| < R2 } where R2 = 1/ lim sup |ak |1/k k→∞

R1 = lim sup |a−k |1/ k . k→∞

9.2 Laurent Expansions

121

If R1 < R2 , D is an annulus and f is analytic in D.

R1

R2

Proof By Theorem 2.8, f1 (z) =

∞

ak z k converges for |z| < R2

k=0

and similarly f 2 (z) =

−1

ak z k =

−∞

converges for

∞

1

1 a−k ( )k z

1 < 1 , or |z| > R1 . z R1

k Hence ∞ −∞ ak z converges for all z in the intersection. Also, since f 1 is a power series and f 2 (z) = g(1/z) where g is a power series, f 1 and f2 are both analytic in their respective domains of convergence. Hence f is analytic in the intersection of these domains. 9.9 Theorem If f is analytic in the annulus A: R1 < |z| < R2 , then f has a Laurent expansion, k f (z) = ∞ k=−∞ ak z , in A. Proof Let C1 and C2 represent circles centered at 0 of radii r1 and r2 , respectively, with R1 < r1 < r2 < R2 . Fix z with r1 < |z| < r2 . Then g(w) =

f (w) − f (z) w−z

122


is analytic in A, and by Cauchy’s Theorem g(w)dw = 0. C 2 −C1

(See Example 2 following Theorem 8.6.) Thus f (z) f (w) dw = dw. (1) C 2 −C1 w − z C 2 −C1 w − z Note then that C2 dw/(w − z) = 2πi , according to Lemma 5.4, while C 1 dw/(w − z) = 0 by Cauchy’s Theorem so that C 2 −C1

f (z) dw = 2πi f (z). w−z

(2)

Combining (1) and (2), we have 1 1 f (w) f (w) f (z) = dw − dw. 2πi C2 w − z 2πi C1 w − z

(3)

Now, on C2 , |w| > |z| so that 1 1 = w−z w 1−

z w

=

z 1 z2 + 2 + 3 + ··· w w w

while on C1 , since |w| < |z|, −1 1 w w2 1 = = − − 2 − 3 − ··· , w−z z−w z z z the convergence being uniform in both cases. Substitution into (3), then, yields ⎛ ⎞

∞ −∞ k

f (w)z k f (w)z 1 1 ⎝ ⎠ dw dw + f (z) = 2πi C2 2πi C1 wk+1 wk+1 k=0

k=−1

and switching the order of summation and integration, f (z) =

∞

k=−∞

ak z k , ak =

1 2πi

C

f (w) dw wk+1

where C is any circle in A centered at 0, for all z ∈ A. For although, in the course of the proof, we have C2 for k ≥ 0 C= C1 for k < 0,


123

in fact C can be taken as any circle in A centered at 0. This follows again from the fact that f (w) g(w) = k+1 w is analytic in A and from the Cauchy Closed Curve Theorem. Note that the Laurent expansion is unique. That is, if ∞

f (z) =

an z n

−∞

in an annulus, then ak = where C is as above. For if C, and thus C

Since

∞

1 2πi

−∞ an z

n

C

f (z) dz z k+1

(4)

converges in A, it converges uniformly along

∞

f (z) dz = an z n−k−1 dz. z k+1 C n=−∞

(5)

2πi p = −1 z dz = 0 any integer p = −1, C p

it follows that

C

f (z) dz = 2πiak , z k+1

proving (4). 9.10 Corollary If f is analytic in the annulus R1 < |z −z 0 | < R2 , then f has a unique representation f (z) =

∞

ak (z − z 0 )k

−∞

where

1 ak = 2πi

C

f (z) dz (z − z 0 )k+1

and C = C(z 0 ; R) with R1 < R < R2 . Proof Simply apply the previous results to g(z) = f (z +z 0 ), which is analytic in an annulus centered at 0.

124


If we set R1 = 0, we obtain: 9.11 Corollary If f has an isolated singularity at z 0 , then for some δ > 0, and 0 < |z − z 0 | < δ f (z) =

∞

ak (z − z 0 )k ,

−∞

where the ak are defined as in Corollary 9.10. E XAMPLES (i) 1 (z + 1)2 = + 2 + z for all z = 0. z z (ii) 1 z 2 (1 −

1 (1 + z + z 2 + · · · ) z) z2 1 1 = 2 + + 1 + z + · · · for 0 < |z| < 1. z z =

(iii) −1 −1 1 = 2 = z 2 (1 − z) z (z − 1) [1 + (z − 1)]2 (z − 1) −1 = + 2 − 3(z − 1) + 4(z − 1)2 − + · · · z−1 for 0 < |z − 1| < 1. (iv) exp(1/z) = 1 +

1 1 + 2 + · · · for z = 0. z 2z ♦

9.12 Definition If f (z) = ak (z − z 0 )k is the Laurent expansion of f about an isolated singularity −1 ∞ k k z0, −∞ ak (z − z 0 ) is called the principal part of f at z 0 ; 0 ak (z − z 0 ) is called the analytic part. Because of the uniqueness of the Laurent expansion, we can derive the following characterizations of the principal parts around the different types of singularities. (i) If f has a removable singularity at z 0 , all the coefficients C−k of its Laurent expansion about z 0 , for k > 0, are 0.


125

Proof Since f (z) = g(z) for z = z 0 , the Laurent expansion for f must agree with the Taylor expansion for g around z 0 .

E XAMPLE

z2 z4 sin z =1− + − +··· . z 3! 5! ♦

(ii) If f has a pole of order k at z 0 , C−k = 0 but C−N = 0 for all N > k. Proof Since f (z) = A(z)/B(z) where A(z 0 ) = 0 and B has a zero of order k at z 0 , f (z) =

Q(z) , (z − z 0 )k

where Q is analytic and nonzero at z 0 . Hence if Q(z) = f (z) =

∞

n=0

an

∞

n=0 an (z

− z 0 )n , then

∞

(z − z 0 )n = C j (z − z 0 ) j (z − z 0 )k j =−k

where C j = a j +k . Thus, C−k = a0 = Q(z 0 ) = 0.

(iii) If f has an essential singularity at z 0 , it must have infinitely many nonzero terms in its principal part. Proof Otherwise (z − z 0 ) N f (z) would be analytic at z 0 for large enough N and f would have a pole at z 0 . The so-called partial fraction decomposition of proper rational functions can be derived as a corollary of the theory of Laurent expansions. 9.13 Partial Fraction Decomposition of Rational Functions Any proper rational function R(z) =

P(z) P(z) = , k 1 Q(z) (z − z 1 ) (z − z 2 )k2 · · · (z − z n )kn

where P and Q are polynomials with deg P < deg Q, can be expanded as a sum of polynomials in 1/(z − z k ), k = 1, 2, . . . , n.

126


Proof Since R has a pole of order at most k1 at z 1 , 1 R(z) = P1 + A1 (z) z − z1 where P1 (1/(z − z 1 )) is the principal part of R around z 1 and A1 is the analytic part. Furthermore 1 A1 (z) = R(z) − P1 z − z1 has a removable singularity at z 1 and the same principal parts as R at z 2 , . . . , z n . Thus, if we take P2 (1/(z − z 2 )) to be the principal part of R around z 2 and proceed inductively, we find 1 1 1 An (z) = R(z) − P1 + P2 + · · · + Pn z − z1 z − z2 z − zn is an entire function (once it is defined “correctly” at z 1 , z 2 , . . . z n ). Furthermore, An is bounded since R and all its principal parts approach 0 as z → ∞. Thus, by Liouville’s Theorem (5.10), An is constant; indeed An ≡ 0. Hence. 1 1 1 + P2 + · · · + Pn . R(z) = P1 z − z1 z − z2 z − zn

Exercises 1. Suppose f (z) → ∞ as z → z 0 , an isolated singularity. Show that f has a pole at z 0 . 2. Does there exist a function f with an isolated singularity at 0 and such that | f (z)| ∼ exp(1/|z|) near z = 0? 3.* Suppose that f is an entire 1 − 1 function. Show that f (z) = az + b. 4. Suppose f is analytic in the punctured plane z = 0 and satisfies | f (z)| ≤ is constant.

√ √ |z| + 1/ |z|. Prove f

5.* Suppose f and g are entire functions with | f (z)| ≤ |g(z)| for all z. Prove that f (z) = cg(z), for some constant c. 6. Verify directly that e1/z takes every value (with a single exception) in the annulus: 0 < |z| < 1. What is the missing value? 7. Suppose f and g have poles of order m and n, respectively, at z 0 . What can be said about the singularity of f + g, f · g, f /g at z 0 ? 8.* Suppose f has an isolated singularity at z 0 . Show that z 0 is an essential singularity if and only if there exist sequences {an } and {βn } with {an } → z 0 , {βn } → z 0 , { f (an )} → 0, and { f (βn )} → ∞.

Exercises

127

9. Classify the singularities of 1 z4 + z2 b. cot z c. csc z exp(1/z 2 ) . d. z−1 a.

10.* Find the principal part of the Laurent expansion of f (z) =

1 (z 2 + 1)2

about the point z = i. 11. Find the Laurent expansion for a.

1 about z = 0 z4 + z2

b.

exp(1/z 2 ) about z = 0 z−1

c.

1 about z = 2. z2 − 4

12.* Find the Laurent expansion of f (z) =

1 (in powers of z) for z(z − 1)(z − 2)

a. 0 < |z| < 1 b. 1 < |z| < 2 c. |z| > 2. 13.* Let {a1 , a2 , ..., ak } be a set of positive integers and R(z) =

1 . (z a1 − 1)(z a2 − 1) · · · (z ak − 1)

Find the coefficient c−k in the Laurent expansion for R(z) about the point z = 1. 14. Show that if f is analytic in z = 0 and “odd” (i.e., f (−z) = − f (z)) then all the even terms in its Laurent expansion about 0 are 0. 15. Find partial fraction decompositions for 1 1 . b. 2 a. 4 z + z2 z +1 16. Suppose f is analytic in a deleted neighborhood D of z 0 except for poles at all points of a sequence {z n } → z 0 . (Note that z 0 is not an isolated singularity.) Show that f (D) is dense in the complex plane. [Hint: Assume, as in the proof of the Casorati-Weierstrass Theorem, that | f (z) − w| > δ and consider g(z) = 1/( f (z) − w).]

128


17. Show that the image of the unit disc minus the origin under f (z) = csc 1/z is dense in the complex plane a. by noting that sin(1/z) has an essential singularity at z = 0, b. by applying Exercise 16 to f (z). 18. Prove that the image of the plane under a nonconstant entire mapping is dense in the plane. [Hint: If f is not a polynomial, consider f (1/z).]

Chapter 10

The Residue Theorem

10.1 Winding Numbers and the Cauchy Residue Theorem We now seek to generalize the Cauchy Closed Curve Theorem (8.6) to functions which have isolated singularities. Note that by 9.10 and 9.11, if γ is a circle surk rounding a single isolated singularity z 0 and f (z) = ∞ −∞ Ck (z − z 0 ) in a deleted neighborhood of z 0 that contains γ , then f = 2πi C−1 . γ

Thus the coefficient C−1 is of special significance in this context. 10.1 Definition k If f (z) = ∞ −∞ Ck (z − z 0 ) in a deleted neighborhood of z 0 , C−1 is called the residue of f at z 0 . We use the notation C−1 = Res( f ; z 0 ). Evaluation of Residues (i) If f has a simple pole at z 0 ; i.e., if f (z) =

A(z) B(z)

where A and B are analytic at z 0 , A(z 0 ) = 0 and B has a simple zero at z 0 , then C−1 = lim (z − z 0 ) f (z) = z→z 0

A(z 0 ) . B (z 0 )

(1)

Proof Since C−1 + C0 + C1 (z − z 0 ) + · · · , z − z0 (z − z 0 ) f (z) = C−1 + C0 (z − z 0 ) + C1 (z − z 0 )2 + · · · , f (z) =

129

130

10 The Residue Theorem

and lim (z − z 0 ) f (z) = C−1 .

z→z 0

The second equality in (1) follows since A(z) lim (z − z 0 ) f (z) = lim (z − z 0 ) z→z 0 B(z) A(z 0 ) B(z) − B(z 0 ) . = = lim A(z) z→z 0 z − z0 B (z 0 )

z→z 0

(ii) If f has a pole of order k at z 0 , C−1 =

( 1 d k−1 ' k (z − z ) f (z) evaluated at z 0 . 0 (k − 1)! dz k−1

Proof Setting f (z) = C−k (z − z 0 )−k + · · · C−1 (z − z 0 )−1 + C0 + C1 (z − z 0 ) + · · · g(z) = (z − z 0 )k f (z) = C−k + · · · C−1 (z − z 0 )k−1 + C0 (z − z 0 )k + · · · d k−1 g(z) = (k − 1)!C−1 + k!C0 (z − z 0 ) + · · · dz k−1 and the equality follows.

(iii) In most cases of higher-order poles, as with essential singularities, the most convenient way to determine the residue is directly from the Laurent expansion. E XAMPLES 1 = 1. i. Res(csc z; 0) = cos 0 1 i 1 ii. Res ; i = 3 = . 4−1 4 z 4i 1 iii. Res 3 ; 0 = 0. z 1 ; 1 = 1, since iv. Res sin z−1 1 1 1 1 = − + − +··· . ♦ z−1 z − 1 3!(z − 1)3 5!(z − 1)5 Winding Number. To evaluate γ f when γ is a general closed curve (and when f may have isolated singularities), we introduce the following concept. sin

10.1 Winding Numbers and the Cauchy Residue Theorem

131

10.2 Definition Suppose that γ is a closed curve and that a ∈ γ . Then 1 dz n(γ , a) = 2πi γ z − a is called the winding number of γ around a. Note that if γ represents the boundary of a circle (traversed counter-clockwise) 1 if a is inside the circle n(γ , a) = 0 if a is outside the circle. The first identity was proven in Lemma 5.4. The second was shown in Example 1 following the Cauchy Closed Curve Theorem. Also, if γ circles the point a k times— i.e., if γ (θ ) = a + r eiθ , 0 ≤ θ ≤ 2kπ—then 1 n(γ , a) = 2πi

2kπ

i dθ = k,

0

which explains the terminology “winding number.” 10.3 Theorem For any closed curve γ and a ∈ γ , n(γ , a) is an integer. Proof Suppose γ is given by z(t), 0 ≤ t ≤ 1, and set s z˙ (t) F(s) = dt, 0 ≤ s ≤ 1. z(t) −a 0 Then, as we saw in defining the logarithm function (Section 8.2) it follows from ˙ F(s) = that

z˙ (s) z(s) − a

(z(s) − a)e−F (s)

is a constant, and setting s = 0, (z(s) − a)e−F (s) = z(0) − a. Hence e F (s) =

z(s) − a z(0) − a

132


and e F (1) =

z(1) − a =1 z(0) − a

since γ is closed; i.e., z(1) = z(0). Thus F(1) = 2πki for some integer k and n(γ , a) =

1 F(1) = k. 2πi

It follows from Definition 10.2 that if we fix γ and let a vary, n(γ , a) is a continuous function of a (as long as a ∈ γ ). Since it is always integer-valued, we conclude that n(γ , a) is constant in the connected components of the complement of γ . Moreover, n(γ , a) → 0 as a → ∞. Hence n(γ , a) = 0 in the unbounded component of γ (the set of points which can be connected to ∞ without intersecting γ ). Some typical examples are illustrated below. n=1

n=2 n=0

n=0

n=1 n=2

n=1 n=1

n = –1 n=0

n=1

The Jordan Curve Theorem asserts that any simple closed curve divides the plane into exactly two components—one bounded, the other unbounded (here the curve need not necessarily be smooth)—so that if γ is such a “Jordan” curve (and is smooth besides), k if a ∈ Bounded Component n(γ , a) = 0 if a ∈ Unbounded Component. The proof of the Jordan Curve Theorem would lead us too far afield. Nevertheless, in all future examples when we deal with simple closed curves, we will be able to verify directly that n(γ , a) = 0 or ±1 for all a ∈ γ . In fact, by choosing the “proper” orientation, we will be able to show that n(γ , a) = 0 or 1 for all a ∈ γ . (The “proper” orientation will be easily recognized as that one for which the bounded component of γ˜ lies to the left of γ .)

10.1 Winding Numbers and the Cauchy Residue Theorem

133

E XAMPLE Let γ be a semicircle traversed counterclockwise. Then 1 if a is inside the semicircle n(γ , a) = 0 if a is outside. The first assertion can be seen by citing the Closed Curve Theorem to show dz dz = z − a z γ C −a C

γ a

γ

C

where C is the completed circle containing z = a. The second follows from the analyticity of 1/(z − a) in a half-plane containing γ but not a.

γ a

♦ To simplify our terminology, we introduce the following definition. 10.4 Definition γ is called a regular closed curve if γ is a simple closed curve with n(γ , a) = 0 or 1 for all a ∈ γ . In that case, we will call {a : n(γ , a) = 1} the inside of γ . The set of points a where n(γ , a) = 0 is called the outside of γ . 10.5 Cauchy’s Residue Theorem Suppose f is analytic in a simply connected domain D except for isolated singularities at z 1 , z 2 , · · · , z m . Let γ be a closed curve not intersecting any of the singularities.

134


Then

f = 2πi

γ

m

n(γ , z k )Res( f ; z k ).

k=1

Proof (Note that since γ is a “general” curve, we cannot replace it by a finite union of “familiar” curves. Instead we proceed as in Section 9.2.) If we subtract the principal parts 1 1 P1 , · · · , Pm z − z1 z − zm from f , the difference

g(z) = f (z) − P1

1 z − z1

− P2

1 z − z2

− · · · − Pm

1 z − zm

(with the appropriate definitions at z 1 , . . . , z m ) is an analytic function in D. Hence, by the Closed Curve Theorem (8.6) g=0 γ

and

γ

Furthermore, for any k,

γ

(z − z k

)−n

f =

m

k=1 γ

Pk

1 z − zk

dz.

(3)

1 = 0, if n = 1 since (z − z k )n

is the derivative of

(z − z k )1−n , 1−n which is analytic along the closed curve γ . Hence if C−1 C−2 1 = Pk + + ··· , z − zk z − zk (z − z k )2 dz 1 dz = C−1 Pk = 2πi n(γ , z k )Res( f ; z k ) z − zk γ γ z − zk

and by (3), the proof is complete.

10.6 Corollary If f is as above, and if γ is a regular closed curve in the domain of analyticity of f , then γ f = 2πi k Res( f ; z k ), where the sum is taken over all the singularities of f inside γ .

10.2 Applications of the Residue Theorem

135

10.2 Applications of the Residue Theorem 10.7 Definition We say f is meromorphic in a domain D if f is analytic there except at isolated poles. 10.8 Theorem Suppose γ is a regular closed curve. If f is meromorphic inside and on γ and contains no zeroes or poles on γ , and if Z = number of zeroes of f inside γ (a zero of order k being counted k times), P = number of poles of f inside γ (again with multiplicity), then 1 f = Z − P. 2πi γ f Proof Note that f / f is analytic except at the zeroes or poles of f . If f has a zero of order k at z = a, that is, if f (z) = (z − a)k g(z) then

with g(z) = 0,

) * f (z) = (z − a)k−1 kg(z) + (z − a)g (z)

has a zero of order k − 1 at z, and k g (z) f (z) = + f (z) z−a g(z) Hence, at each zero of f of order k, f / f has a simple pole with residue k. Similarly, if f (z) = (z − a)−k g(z), then

−k g (z) f (z) = + , f (z) z−a g(z)

so that at each pole of f, f / f has a simple pole with residue −k. By Corollary 10.6, then

1 f f = Res = Z − P. 2πi γ f f

136


If we take f to be analytic, we obtain 10.9 Corollary (Argument Principle) If f is analytic inside and on a regular closed curve γ (and is nonzero on γ ) then 1 f Z( f ) = the number o f zer oes o f f i nsi de γ = . 2πi γ f Remarks 1. The above is known as the “Argument Principle” because if γ is given by z(t), 0 ≤ t ≤ 1, 1 log f (z(1)) − log f (z(0)) 1 f = = Arg f (z) (1) 2πi γ f 2πi 2π as z travels around γ from the starting point z(0) to the terminal point z(1) = z(0). To prove (1), we split γ into a finite number of simple arcs γ1 : z(t), 0 ≤ t ≤ t1 γ2 : z(t), t1 ≤ t ≤ t2 ··· γn : z(t), tn−1 ≤ t ≤ tn = 1.

γ3

γn

γ1

γ2

Since an analytic branch of log f can be defined in a simply connected domain containing γ1 , f = log f (z(t1 )) − log f (z(0)). γ1 f Similarly γk

f = log f (z(tk )) − log f (z(tk−1 )), f

k = 2, 3, . . . , n.


We note that

137

γ

=

+

γ1

γ2

+···+

γn

,

and the first equality in (1) follows. Note, also, that since z(0) = z(1) and since log w = log |w| + i Arg ω, ) * log f (z(1)) − log f (z(0)) = i Arg f (z(1)) − Arg f (z(0)) , and the second equality follows. 2. We may also view γ f / f as the winding number of the curve f (γ (z)) around z = 0. (See Definition 10.2.) Thus, if f is analytic inside and on γ , the number of zeroes of f inside γ is equal to the number of times that the curve f (γ ) winds around the origin. By considering f (z) − a, it follows that the number of times that f = a inside γ equals the number of times that f (γ ) winds around the complex number a. As an example, consider the function described in Exercise 3b of Chapter 7. 10.10 Rouché’s Theorem Suppose that f and g are analytic inside and on a regular closed curve γ and that | f (z)| > |g(z)| for all z ∈ γ . Then Z( f + g) = Z( f )

i nsi de γ .

Proof Note first that if f (z) = A(z)B(z) f A B = + f A B

so that

γ

f = f

γ

A + A

γ

B . B

Thus, if we write f +g = f

g 1+ , f

g)

f

1 1 (f + = + f + g 2πi f 2πi γ γ 1+ g 1 f = Z( f ) + . 2πi γ 1 + gf

Z( f + g) =

1 2πi

γ

1+ 1+

g f g f

138


But this last integral is zero since, by hypothesis, (1 + g/ f )(γ ) remains within a disc of radius 1 around z = 1. Hence the winding number of (1 + g/ f )(γ ) around 0 is 0 [i.e., setting ω = 1+ g/ f it follows that ω(z) remains in the right half-plane for z ∈ γ and hence that γ ∗ dω ω = 0.]

(1 + gf ) (γ )

0

1

2

E XAMPLE Since |4z 2 | > |2z 10 + 1| on |z| = 1, each of the polynomials 2z 10 + 4z 2 + 1 and 2z 10 − 4z 2 + 1 has exactly two zeroes in |z| < 1.

♦

Recall that according to the Cauchy Integral Formula (6.4) 1 f (ω) dω f (z) = 2πi C ω − z where C is a circle containing z. By application of the Residue Theorem, we can extend the result as follows. 10.11 Generalized Cauchy Integral Formula Suppose that f is analytic in a simply connected domain D and that γ is a regular closed curve contained in D. Then for each z inside γ and k = 0, 1, 2 . . ., k! f (ω) f (k) (z) = dω. 2πi γ (ω − z)k+1 Proof Note that since f (ω) = f (z) + f (z)(ω − z) + · · · +

f (k) (z) (ω − z)k + · · · k!


139

throughout a neighborhood of z, Res

f (k) (z) f (ω) . ; z = k! (ω − z)k+1

Since f (ω)/(ω − z)k+1 has no other singularities in D, the result follows from Corollary 10.6. We now derive an extension of Theorem 7.6 for the limit of analytic functions. 10.12 Theorem Suppose a sequence of functions fn , analytic in a region D, converges to f uniformly on compacta of D. Then f is analytic, f n → f in D and the convergence of f n is also uniform on compacta of D. Proof We proved the analyticity of f in Theorem 7.6. By the Integral Formula 10.11, if we pick any z 0 ∈ D and let C = C(z 0 ; r ) for some r < 1, 1 fn (ω) − f (ω) f n (z) − f (z) = dω 2πi C (ω − z)2 for all z in D(z 0 ; r ). Moreover, if we take n large enough so that | f n − f | < r 2 /4 throughout the compact D(z 0 ; r ), it follows that | f n (z) − f (z)| < for all z in D(z 0 ; r/2). Thus, to see that the convergence is uniform on compacta, we need only note that any compact subset D can be covered by finitely many discs of the form: |z − z 0 | < r/2. 10.13 Hurwitz’s Theorem Let { f n } be a sequence of non-vanishing analytic functions in a region D and suppose f n → f uniformly on compacta of D. Then either f ≡ 0 in D or f (z) = 0 for all z ∈ D. Proof Suppose f (z) = 0 for some z ∈ D. If f ≡ 0, there is some circle C centered at z and such that f (z) = 0 on C; hence fn f → fn f

uniformly on C.

140

However

while


1 2πi 1 2πi

C

f = Z( f ) ≥ 1, f

C

f n = Z( f n ) = 0. fn

Hence f ≡ 0.

[Note that it is possible to have f ≡ 0 despite the fact that f n (z) = 0 for all n. Consider, for example, f n (z) = (1/n)e z .] E XAMPLE Since sin π = 0, there must be some n 0 such that z−

z3 z5 z 2n+1 + − +··· 3! 5! (2n + 1)!

has a zero in |z − π| < 1 for all n > n 0 .

♦

10.14 Corollary Suppose that f n is a sequence of analytic functions in a region D, that f n → f uniformly on compacta in D, and that f n = a. Then either f ≡ a or f = a in D. Proof Consider gn (z) = f n (z) − a, etc.

10.15 Theorem Suppose that f n is a sequence of analytic functions, and that f n → f uniformly on compacta in a region D. If f n is 1-1 in D for all n, then either f is constant or f is 1-1 in D. Proof Assume z 1 = z 2 , f (z 1 ) = f (z 2 ) = a and take disjoint discs D1 and D2 (in D) surrounding z 1 and z 2 , respectively. If f ≡ a, by 10.13, f n (z) = a must have a solution in D1 once n is large enough. (Otherwise we could find a subsequence f nk → f with no a-values D1 .) But then since fn is 1-1, fn (z) = a throughout D2 for all large n and hence f (z 2 ) = a, contradicting our hypothesis.

Exercises

141

Exercises 1. Determine the singularities and associated residues of 1 a. 4 b. cot z z + z2 exp(1/z 2 ) c. csc z d. z−1 1 1 f. sin e. 2 z z + 3z + 2 1 g. ze3/z , a = 0. h. 2 az + bz + c 2. Use the Residue Theorem to evaluate dz a. |z|=1 cot z dz b. |z|=2 (z − 4)(z 3 − 1) 1 c. |z|=1 sin dz z

d. |z|=2 ze3/z dz.

3. Prove that for any positive integer n, Res((1 − e−z )−n ; 0) = 1. [Hint: Consider dz −z n C (1 − e ) and make the change of variables ω = 1 − e−z to show 1 Res((1 − e−z )−n ; 0) = Res ; 0 .] ωn (1 − ω) 4.* Show that |z|=1 (z + 1/z)2m+1 dz = 2π i 2m+1 m , for any nonnegative integer m. 5.* Let C be a regular curve enclosing the distinct points ω1 , ω2 , ...ωn and let p(ω) = (ω − ω1 )(ω − ω2 ) · · · (ω − ωn ). Suppose that f (ω) is analytic in a region that includes C. Show that P(z) =

1 f (ω) p(ω) − p(z) · dω 2π i C p(ω) ω−z

is a polynomial of degree n − 1, with P(ωi ) = f (ωi ), i = 1, 2, ...n.

6. Suppose f is defined by f (z) =

γ

φ(ω)dω , ω−z

where γ is a compact curve, φ is continuous on γ , and z ∈ γ . Show that φ(ω)dω f (z) = γ (ω − z)2 directly by considering

f (z + h) − f (z) . h Give an alternate proof of Theorem 10.11. 7. Suppose that f is entire and that f (z) is real if and only if z is real. Use the Argument Principle to show that f can have at most one zero. (Compare this with Exercise 13 of Chapter 5.) 8.* a. Show that Rouche’s Theorem remains valid if the condition: | f | > |g| on γ is replaced by: | f | ≥ |g| and f + g = 0 on γ . b. Find the number of zeroes of z 5 + 2z 4 + 1 in the unit disc.

142


9. Find the number of zeroes of a. f 1 (z) = 3ez − z in |z| ≤ 1 1 b. f 2 (z) = ez − z in |z| ≤ 1 3 c. f 3 (z) = z 4 − 5z + 1 in 1 ≤ |z| ≤ 2 d. f 4 (z) = z 6 − 5z 4 + 3z 2 − 1 in |z| ≤ 1. 10.* Suppose λ > 1. Show that λ − z − e−z = 0 has exactly one root (which is a real number) in the right half-plane. 11. Suppose f is analytic inside and on a regular closed curve γ and has no zeroes on γ . Show that if m is a positive integer then

f (z) 1 dz = zm (z k )m 2π i γ f (z) k

where the sum is taken over all the zeroes of f inside γ . 12. Show that for each R > 0, if n is large enough, Pn (z) = 1 + z +

zn z2 + ··· + has no zeroes in |z| ≤ R. 2! n!

13.* a. Let P(z) be any polynomial of the form: a0 + a1 z + a2 z 2 + · · · + an z n , with all ai real and 0 ≤ a0 ≤ a1 ≤ · · · ≤ an . Prove that all the zeroes of P(z) lie inside the unit disc by applying Rouche’s Theorem to (1 − z)P(z). b. Prove that, for any ρ < 1, the polynomial Pn (z) = 1 + 2z + 3z 2 + · · · + (n + 1)z n has no zeroes inside the circle |z| < ρ if n is sufficiently large. 14. Derive the Fundamental Theorem of Algebra as a corollary of Rouché’s Theorem. 15. Supply the details of the following proof of Rouché’s Theorem (due to Carathéodory). Set J (λ) =

( f + λg) 1 , 2π i γ f + λg

0 ≤ λ ≤ 1.

Note that J (λ) is defined for all λ, 0 ≤ λ ≤ 1. Furthermore J (λ) is a continuous function of λ and is always integer-valued. Hence J is constant; in particular, J (0) = J (1) so that Z( f ) = Z( f + g). 16. Recall, as in 8.2, that

z log(z 2 − 1) = √

2ζ

2 ζ2 − 1

dζ

is analytic in the plane minus the interval (−∞, 1]. Hence, so is

z 2 − 1 = exp

1 log(z 2 − 1) . 2

(1)

in the entire plane minus the interval [−1, 1]. [Hint: Show that z 2 − 1 (as defined in (1)) is analytic Use the Argument Principle to show that z 2 − 1 is continuous along the interval (−∞, −1) and then apply Morera’s Theorem.] √ 17. Show that an analytic 3 (z − 1)(z − 2)(z − 3) can be defined in the entire plane minus [1, 3].

Chapter 11

Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Introduction In the next section, we will see how various types of (real) definite integrals can be associated with integrals around closed curves in the complex plane, so that the Residue Theorem will become a handy tool for definite integration.

11.1 Evaluation of Definite Integrals by Contour Integral Techniques ∞ I Integrals of the Form −∞ ( P(x)/ Q(x))d x, where P and Q are polynomials. From real-variable calculus we know that an integral of this type will converge if Q(x) = 0 and deg Q − deg P ≥ 2. Making these assumptions, we note that

∞ −∞

P(x) d x = lim R→∞ Q(x)

R

−R

P(x) d x, Q(x)

and we seek to estimate the second integral for large values of R. Let C R be the closed contour consisting of the real line segment from −R to R and the upper semi-circle R centered at the origin and of radius R large enough to enclose all zeroes of Q lying in the upper half-plane.

iR CR

–R

R

143

144

11 Applications of the Residue Theorem to the Evaluation of Integrals and Sums

By the Residue Theorem

P(z) P dz = 2πi ; zk Res Q C R Q(z) k

where the points z k are the zeroes of Q in the upper half-plane. Thus R

P(x) P(z) P dx + dz = 2πi ; zk (1) Res Q −R Q(x) R Q(z) k To estimate R P/Q, note that since deg Q − deg P ≥ 2, by the usual M − L estimates A P π·R· 2 R R Q

and hence lim

R→∞ R

P(z) dz = 0. Q(z)

(2)

Combining (1) and (2) shows that ∞

P(x) P d x = 2πi ; zk Res Q −∞ Q(x) k

E XAMPLE

∞ −∞

2

dx 1 = 2πi ; z Res k x4 + 1 z4 + 1 k=1

where z 1 = eiπ/4 and z 2 = e3πi/4 represent the poles of 1/(z 4 + 1) in the upper half-plane. Since each is a simple pole, the residues are given by the values of 1/4z 3 at the poles. Thus √ 1 √ 1 −z 1 1 iπ/4 = − ( 2 + i 2) ;e Res 4 = 3 = 4 8 z +1 4z 1

and Res

√ 1 √ 1 i3π/4 ; e = ( 2 − i 2), 4 8 z +1

√ π 2 dx . ♦ = 4 2 −∞ x + 1 ∞ ∞ II. Integrals of the Form −∞ R(x) cos x d x or −∞ R(x) sin x d x. Assuming that P(x) R(x) = Q(x) so that

∞

where P and Q are polynomials and Q(x) = 0 (except perhaps at a zero of cos x or sin x), the above integrals converge as long as deg Q > deg P.

11.1 Evaluation of Definite Integrals by Contour Integral Techniques

145

Integrating R(z) cos z along the same contour as in Type I is not appropriate since lim R(z) cos z dz = 0. M→∞ M

If we consider

R(z)eiz dz, CM

however, we will be able to show that R(z)eiz dz → 0 M

so that

∞

iz

R(z)e dz →

∞ −∞

−∞

CM

R(x) cos x d x and

R(x)ei x d x. ∞

−∞

(3)

R(x) sin x d x

can then be determined as the real and imaginary parts of the limit in (3). Hence, applying the Residue Theorem in (3), we see that ∞

iz R(x) cos x d x = Re 2πi Res(R(z)e ; z k ) ∞

−∞

k

and −∞ R(x) sin x d x = Im [2πi k Res(R(z)eiz ; z k )], where the points z k are the poles of R(z) in the upper half-plane. To show that two subsets:

M

R(z)eiz dz → 0, and complete the argument, we split M into A = {z ∈ M : Im z ≥ h} B = {z ∈ M : Im z < h}.

iM A A

B –M

ih

B M

146


Using the facts that R(z) K /|z| and |e z | = eRe z , we obtain R(z)eiz dz K A

e−h · π M = C1 e−h . M

But

R(z)eiz dz B

so

If we now choose h =

√

M

K h 4h = C2 , M M

R(z)eiz dz C1 e−h + C2

h . M

M, for example, we find

√

M

R(z)eiz dz C1 e−

M

C2 +√ M

and lim

M→∞ M

R(z)eiz dz = 0.

E XAMPLE ∞ To evaluate −∞ (sin x/x)d x, we might write

∞ −∞

sin x d x = Im x

∞ −∞

ei x d x. x

The pole of ei x /x at x = 0 forces us to modify the technique slightly; we write instead: ∞ ∞ ix sin x e −1 d x = Im d x. x −∞ x −∞ Note that

CM

eiz − 1 dz = z

M −M

ei x − 1 dx + x

M

eiz − 1 dz; z

while, according to Cauchy’s Theorem, CM

eiz − 1 dz = 0 z

since the integrand has no poles! Thus

M −M

ei x − 1 dx = x

1 − eiz dz = z M eiz = πi − dz. M z

M

1 dz − z

M

eiz dz z


Since

M (e

iz /z)dz

147

approaches 0 as M → ∞,

∞ −∞

and

ei x − 1 d x = πi x

∞ −∞

sin x d x = π. x

♦

∞ III (A) Integrals of the Form 0 ( P(x)/ Q(x))d x As in I, to insure convergence of the integral, we assume that deg Q − deg P ≥ 2 and that Q(x) = 0 for x ≥ 0. Of course, if the integrand is an even function it can be evaluated as 1 ∞ 2 −∞ (P(x)/Q(x))d x. In other cases, set R(z) = P(z)/Q(z) and consider the integral of log z · R(z) around the keyhole-shaped contour K ,M consisting of √ i. the horizontal line segment I1 from i to M 2 − 2 + i ; ii. the circular arc C M of radius M traced counterclockwise from M 2 − 2 + i to M 2 − 2 − i ; iM CM

K ∋ ,M

–M

C∋

i∋

I1

– i∋

I2

iii. the horizontal line segment I2 from M 2 − 2 − i to − i ; iv. the semi-circle C of radius traced clockwise from −i to i . The inside of K ,M is a simply connected domain not containing 0 and hence log z may be defined there as an analytic function. (For simplicity, we choose 0 < Arg z < 2π.) By the Residue Theorem

lim R(z) log zdz = 2πi Res(R(z) log z; z k ). (4) →0 M→∞

K ,M

k

148


Moreover, assuming is small enough and M large enough so that all the zeroes of ∞ Q lie inside K ,M , the contour integral is related to 0 R(x) d x as follows: i. C R(z) log z dz π maxC |R(z) log z| A| log | since R is continuous at 0 and | log z| < log |z| + 2π. Thus lim R(z) log z dz = 0. ii.

→0 C

R(z) log z dz 2πM · maxC M | log z||R(z)| ≤ AM log M/M 2 since R(z) B/|z|2, and thus lim R(z) log z dz = 0. CM

iii. lim

→0 M→∞

M→∞ C M

I1

R(z) log z dz =

∞ 0

R(x) log x d x and

→0 M→∞

∞

R(z) log z dz = −

lim

R(x)(log x + 2πi )d x.

0

I2

Combining all of the above results we find R(z) log z dz = −2πi lim →0 M→∞

so that by (4)

R(x)d x,

0

K ,M ∞

∞

R(x)d x = −

0

Res(R(z) log z; z k )

k

where the sum is taken over all the poles of R. E XAMPLE ∞ To evaluate 0 d x/(1 + x 3 ), note that at z 1 = eiπ/3 , √ iπ 1 3 log z +i ; Res ; z1 = − 3 1+z 9 2 2 at z 2 = −1 = eiπ ,

and at z 3 = ei5π/3 ,

iπ log z ; ; z2 = Res 3 1+z 3

√ −5πi 1 3 log z −i ; ; z3 = Res 9 2 2 1 + z3

so that

k

log z Res ; zk 1 + z3

=−

2π √ 3 9


and

∞

0

149

2 √ dx π 3. = 1 + x3 9 ♦

(B) Integrals of the form by considering

∞ a

(P(x)/Q(x))d x can be evaluated in a similar manner

log(z − a) CM

P(z) dz Q(z)

CM

a

–M

M

0

with C M as indicated. In fact, since ∞ 0

∞

− a

a

=

,

0

this method can be used to find indefinite integrals of rational functions. (C) Integration around the “keyhole” contour can also be used to evaluate integrals of the form ∞ α−1 x dx P(x) 0 where 0 < α < 1 and P is a polynomial with deg P ≥ 1. Throughout the inside of the contour K ,M , z α−1 = exp [(α − 1) log z] can be defined as an analytic function (again, with 0 < Arg z < 2π, for example). As we integrate along I1 (as → 0) z α−1 = exp((α − 1) log x) = x α−1 while, throughout I2 z α−1 = e(α−1)(log x+2πi) = x α−1 e2πi(α−1) .

150


Since the integrals along the two circular segments approach zero as before, the integral around K ,M is given by the integrals along I1 and I2 and hence

' ( ∞ x α−1 α−1

z 1 − e2πi(α−1) d x = 2πi ; zk , Res P(x) P(z) 0 k

the sum being taken over the zeroes of P. E XAMPLE √ ∞ To evaluate 0 d x/ x(1 + x), note that 1 ; −1 = −i Res √ z(1 + z) and

1 − e−πi

∞

0

so that

∞

√

0

√

x = 2π x(1 + x)

dx = π. x(1 + x)

♦

2π IV 0 R(cos θ, sin θ )dθ where R Represents a Rational Function Here we take a slightly different point of view. Previously, we viewed the definite integrals as integrals along real line segments which were then supplemented into closed contours in the complex plane. In this case, we think of the real integral itself as the parametric representation of a line integral taken around the unit circle. Recall that f (z)dz |z|=1

becomes

2π

f (eiθ )i eiθ dθ

0

on setting z = eiθ , 0 ≤ θ ≤ 2π. 2π More specifically, the integral 0 R(cos θ, sin θ )dθ is equal to

z + 1z z − 1z dz , R 2 2i iz |z|=1 since with z = eiθ dz , iz eiθ + e−iθ 1 1 cos θ = = z+ , 2 2 z dθ =

(5)

11.2 Application of Contour Integral Methods to Evaluationand Estimation of Sums

and sin θ =

eiθ − e−iθ 1 = 2i 2i

z−

151

1 . z

The contour integral (5), as always, can be evaluated by the Residue Theorem. E XAMPLE 0

2π

2 dθ = 2 + cos θ i

dz 2 + 4z + 1 z |z|=1 √ 1 ; 3−2 = 4πRes 2 z + 4z + 1 2 √ = π 3. 3

♦

11.2 Application of Contour Integral Methods to Evaluation and Estimation of Sums I To evaluate sums of the form ∞ n=−∞ f (n), we seek a function g whose residues are given by { f (n) : n = 0, ±1, ±2, . . .}. Suppose we set g(z) = f (z)ϕ(z). Then the function ϕ should have a simple pole with residue 1 at every integer. Such a function is given by ϕ(z) = π

cos π z , sin π z

since sin π z has a simple zero at every integer and π cos πn π cos π z ;n = = 1. Res sin π z π cos πn (Note that sin z has no other zeroes in the complex plane.) We first apply the Residue Theorem to the integral f (z) · π cot π zdz

(1)

CN

where C N is a simple closed contour enclosing the integers n = 0, ±1, ±2, . . . , ±N and the poles of f (which we assume to be finite in number). Thus ⎡ ⎤ N

⎢ ⎥ π f (z) cot π zdz = 2πi ⎣ f (n) + Res( f (z)π cot π z; z k )⎦ (2) CN

where {z k } are the poles of f .

n=−N n=z k

k

152


∞ Furthermore, to insure convergence of n=−∞ f (n), we will assume that | f (z)| ≤ zA2 so that lim z f (z) = 0, (3) z→∞

and by a proper choice of C N , we will be able to show that lim f (z)π cot π z dz = 0. N→∞ C N

Then by (2)

∞

f (n) = −

n=−∞ n=z k

Res( f (z)π cot π z; z k ).

(4)

(5)

k

To demonstrate the existence of a contour C N satisfying (4), we will let C N be the square with vertices ±(N + 12 ) ± (N + 12 )i . Having thus avoided the poles of cot π z, we can show that | cot π z| < 2 on C N . For example, if Re z = x = N + 12 and Im z = y then eπi−2π y + 1 e2πiz + 1 = i πi−2π y cot π z = i 2πiz e −1 e −1 i (N + –¹² )

CN N

N+1

–i (N + –¹²) CN

and

1 − e−2π y | cot π z| = < 1. 1 + e−2π y

Similarly, if Im z = y = N +

1 2

| cot π z| ≤

1 + e−π(2N+1) 1. i. 02π 5 + 3 cos x a + cos x 2. Evaluate ∞ sin2 x d x. x2 0 [Hint: Integrate (e2iz − 1 − 2iz)/z 2 around a large semi-circle.] 3. Evaluate

∞ 0

dx 1 + xn

where n ≥ 2 is a positive integer. [Hint: Consider the following contour.]

e π i/n 2π /n

Exercises

159

4.* Evaluate: ∞ cos ax d x; a ≥ 0 a. (x 2 + 1)2 0 ∞ x2 b. d x (See the hint for exercise 3.) x 10 + 1 0 c.

2π

iθ

ee dθ

0

5.* Show that

2π 0

2π 2m ( cos x)2m d x = m 4 m

for any positive integer m. ∞

6.* Show that

dx

−∞ (1 + x 2 )n+1

=

1 · 3 · 5 · · · (2n − 1)π 2 · 4 · 6 · · · 2n

2 7. a. Show that eiz dz → 0 as R → ∞ where R is the circular segment: z = Reiθ , 0 ≤ θ ≤ π/4. ∞R b. Evaluate 0 cos x 2 d x, 0∞ sin x 2 d x. Note: The convergence of the above integrals can be proven for example by making the substitution u = x 2 and applying Dirichlet’s Test. 8. Suppose f is a rational function of the form P/Q with deg Q − deg P ≥ 2. Show that the sum of the residues of f is zero. 9. Evaluate a. b. c.

∞

n=1 ∞

n=1 ∞

n=1

1 , n2 + 1 1 , n4 (−1)n . n2 + 1

10.* a. Show that

1

Cn z sin π z

dz → 0 as N → ∞, where C N is the square with vertices ±(N + 12 ) ±

(N + 12 )i. (See Chapter 3, exercise 16.) 1 1 1 π 1 around a suitable contour, show that 1 − + − + · · · = . b. By integrating (2z − 1) sin π z 3 5 7 4 11.* Show that

∞

ekx x dx −∞ 1 + e

converges if 0 < k < 1. Find its value by integrating around the rectangle with vertices at ±R and at ±R + 2π i. 12.* Suppose f is analytic for |z| ≤ 1, and let log z be defined so that Im log z = arg z ∈ [ 0, 2π ). Prove 1 f (z) log z dz = 01 f (x)d x that 2π i |z|=1 13. Evaluate

∞

1 3n . n 8n

n=0

160


14. Show that

∞

2n n=0

n

1 1 xn = √ as long as|x| < . 4 1 − 4x

Note: This is the sum of the middle column in Pascal’s Triangle for powers of 1 + x.

1 x

1

4x

1

.

x3

3x 2

3x

1

x2

2x

1

x4

4x3

6x2 . . .

.

The equation can also be verified by applying the binomial expansion for (1 − 4x)−1/2 . √ 15. Complete Example 3 of Section 2-III by showing |(z − 1)2 (z + 1)| ≤ (16/9) 3 throughout |z| = 1. 2 2 2 [Hint: Maximize a b given a + b = 4.]

z | z + 1|

| z – 1|

–1

16. a. Show that

1

(z − 1)2 (z + 1) √ √ ≤ 2 2 for |z| = 2 z2

and thereby obtain an improved estimate for the example cited in (15). b. Show that (z − 1)2 (z + 1) √ max ≥ 2 2 for any R > 0. |z|=R z2 (Thus, in a sense, the estimate in (a) is the best possible.) 17.* a. Express

n

(−1)k

k=0

3n k

as a contour integral.

n b. Use the integral above to prove that

k=0

n k

(−1)k

3n k

n ≤ 4n . k

Chapter 12

Further Contour Integral Techniques

12.1 Shifting the Contour of Integration We have already seen how the Residue Theorem can be used to evaluate real line integrals. The techniques involved, however, are in no way limited to real integrals. To evaluate an integral along any contour, we can always switch to a more “convenient” contour as long as we account for the pertinent residues of the integrand. E XAMPLE 1 Consider

I

e z dz (z + 2)3

where I is the line z(t) = 1 + i t, −∞ < t < ∞. Let C R be the left semicircle of radius R > 3 centered at z = 1. Then 1+i R e z dz e z dz ez + = 2πi Res ; −2 . 3 3 (z + 2)3 1−i R (z + 2) C R (z + 2) Since e z is bounded by e in the lefl half-plane x ≤ 1, as R → ∞ e z dz →0 3 C R (z + 2) and

I

e z dz ez = 2πi Res ; −2 . (z + 2)3 (z + 2)3

161

162

12 Further Contour Integral Techniques

1 + iR CR

1

1 – iR

To evaluate the residue, we write z

e =e

−2 z+2

e

=e

−2

(z + 2)2 1 + (z + 2) + + ··· 2

so that

1 ez Res ; −2 = 2 3 (z + 2) 2e and I

πi e z dz = 2. (z + 2)3 e ♦

E XAMPLE 2 Evaluate

dz √ 2 6z − 5z + 1 |z|=1 √ where the square root is the positive 2 at the point z = 1. Recall (see Exercise 10–16) that since 6z 2 − 5z + 1 has its zeroes at z = 12 and √ z = 13 , 6z 2 − 5z + 1 is analytic in the plane minus the interval 13 ≤ z ≤ 12 . √ To evaluate the√integral, we switch to the contour |z| = R. Then, since 6z 2 − 5z + 1 ∼ 6 z for large z, it follows that 2πi dz dz 1 = √ . √ → √ 2 z 6 |z|=R 6 6z − 5z + 1 |z|=R

12.1 Shifting the Contour of Integration

163

|z| = R |z| = 1 –¹ –¹ ³ ²

1

R

To formally justify the last step, suppose (in general) that f (z) = z + (z), where (z)/z → 0 as z → ∞. Then 1 dz (z) dz − =− dz f (z) z z(z + (z)) |z|=R |z|=R |z|=R (z) → 0 as R → ∞. ♦ 2π max |z|=R z + (z) E XAMPLE 3 Based on numerical evidence, it was conjectured that n

n k (−1) → 0 as n → ∞. k k=0

A proof of the conjecture can be given as follows: Note that sin π z f (z) = π z(1 − z)(1 − z/2) . . . (1 − z/n) satisfies f (k) =

n k

for any nonnegative integer k.

√ f (z) (taken as positive at the Because f (z) is zero-free in −1 < Re z < n + 1, origin) is analytic there. By the Residue Theorem, then n

1 π n k = dz (1) (−1) f (z) k 2πi C sin π z k=0

where C is any contour in −1 < Re z < n + 1 which winds once about each integer 0, 1, . . . , n and never about any other integer.

164


Suppose we let C = C M be the rectangle formed by the lines Re z = −1/2, Re z = n + 1/2 and Im z = ±M. Then √ π πdz dz = f (z) √ sin π z z(1 − z)(1 − z/2) . . . (1 − z/n) sin π z CM CM and letting M → ∞, we conclude n+1/2+i∞ n −1/2−i∞

π 1 n dz . (−1)k + f (z) = k 2πi sin π z −1/2+i∞ n+1/2−i∞ k=0

Since the integrand is invariant (aside from a ± sign) under the substitution z → n−z, we need only estimate the first integral. Now, when Re z = − 12 , 1 1 1 1 1+ 1+ ... 1 + |z(1 − z)(1 − z/2) . . . (1 − z/n)| ≥ 2 2 4 2n 1 1 1 1 1√ 1 + 1 1 + ... 1 + ≥ 2 2 n √ n+1 = 2 and so the first integral is bounded by dz 1 ≤ √A . √ √ √ 4 4 n 2π n + 1 Re z=−1/2 sin π z Hence

n

n k (−1) → 0 as n → ∞. k k=0

♦

12.2 An Entire Function Bounded in Every Direction Recall that, according to Liouville’s Theorem, every nonconstant entire function is unbounded. Nevertheless, one may wonder whether there is a nonconstant entire function which is bounded along every ray from the origin. The answer to this question is yes! However, there seems to be no way of describing such a function in closed form. Instead, the function will be given in integral form and the crucial estimate will then be obtained by switching the contour of integration. The strategy is as follows: We will find a nonconstant entire function f which is bounded by 1 outside of the strip |Im z| ≤ π. If we consider g(z) = f (z − 2πi ),

12.2 An Entire Function Bounded in Every Direction

165

g y > y − θ , zt e ≤ exp −[(y − θ )R] ≤ e−cR tt

and

CR

π e zt dt R · e−cR , tt 2

which approaches 0 as R → ∞. We note that for every nonconstant entire function, there is always some polygonal line along which the function is not only unbounded but actually approaches infinity. We prove this result in Chapter 15. (See Exercise 6.)

Exercises

1.* a. Evaluate

ez I (z + 1)4

dz,

where I is the imaginary axis (from −i∞ to +i∞). b. Evaluate 1+i∞ z a dz, 0 < a < ∞. 1−i∞ z 2

2.* Evaluate

|z|=2

dz . 2 4z − 8z + 3

3. Evaluate γ ez log zdz where log z is that branch for which log 1 = 0 and γ is the parabola: γ (t) = 1 − t 2 + it, − ∞ < t < ∞. 4. Show that

n

k=0

(−1)k

1/3 n →0 k

as n → ∞.

168


5. a. Obtain an improved estimate for n

k=0

(−1)k

n k

by integrating along the lines Re z = −3/4 and Re z = n + 3/4. b. Estimate n

n k (−1) k k=0

by integrating along Re z = −1 + δ and Re z = n + 1 − δ. Find an optimal δ. [Note: Numerical evidence suggests n

1 n k ∼ √ for even n]. (−1) k n k=0

6. * Suppose g is the entire function (bounded on every ray) described in the last section. Show that g(x + 2πi) → ∞ as x → ∞.

Chapter 13

Introduction to Conformal Mapping

In this chapter, we take a closer look at the mapping properties of an analytic function. Throughout the chapter, all curves z(t) are assumed to be such that z˙ (t) = 0 for all t.

13.1 Conformal Equivalence 13.1 Definition Suppose two smooth curves C1 and C2 intersect at z 0 . Then the angle from C1 to C2 at z 0 , ∠C1 , C2 , is defined as the angle measured counterclockwise from the tangent line of C1 at z 0 to the tangent line of C2 at z 0 .

C2

θ

C1

z0

13.2 Definition Suppose f is defined in a neighborhood of z 0 . f is said to be conformal at z 0 if f preserves angles there. That is, for each pair of smooth curves C1 and C2 intersecting

169

170

13 Introduction to Conformal Mapping

at z 0 , ∠C1 , C2 = ∠1 , 2 where 1 = f (C1 ), 2 = f (C2 ). Similarly, we say f is conformal in a region D if f is conformal at all points z ∈ D. Note that f (z) = z 2 is not conformal at z = 0. For example, the positive real axis and the positive imaginary axis are mapped onto the positive real axis and negative real axis, respectively. However, as we shall see below, it is conformal at all other points of the complex plane. 13.3 Definition a. f is locally 1-1 at z 0 if for some δ > 0 and any distinct z 1 , z 2 ∈ D(z 0 ; δ), f (z 1 ) = f (z 2 ). b. f is locally 1-1 throughout a region D if f is locally 1-1 at every z ∈ D. c. f is a 1-1 function in a region D, if for every distinct z 1 , z 2 ∈ D, f (z 1 ) = f (z 2 ). Again, note that f (z) = z 2 is not locally 1-1 at z = 0 since f (z) = f (−z) for all z. However, f is locally 1-1 at all points z = 0 (see Exercise 1). 13.4 Theorem Suppose f is analytic at z 0 and f (z 0 ) = 0. Then f is conformal and locally 1-1 at z 0 . Proof (Conformality) Let C : z(t) = x(t) + i y(t) be a smooth curve with z(t0 ) = z 0 . Then the tangent line to C at z 0 has the direction of z˙ (t0 ) = x (t0 ) + i y (t0 ) so that its angle of inclination with the positive real axis is Arg z˙ (t0 ). If we set = f (C), then is given by ω(t) = f (z(t)) and the angle of inclination of its tangent line at f (z 0 ) is equal to Arg ω(t ˙ 0 ) = Arg

)

* f (z 0 )˙z (t0 ) = Arg f (z 0 ) + Arg z˙ (t0 ).

Hence the function f maps all curves at z 0 in such a way that the angles of inclination are increased by the constant Arg f (z 0 ). Thus, if C1 and C2 meet at z 0 and 1 , 2 are their respectively images under f , it follows that ∠1 , 2 = ∠C1 , C2 . To show f is 1-1 in a neighborhood of z 0 , let f (z 0 ) = α and take δ > 0 small enough so that f (z) − α has no other zeroes in D(z 0 ; δ ). Such a δ can always be found for otherwise we would have f (z 0 ) = 0 (Theorem 6.10). If we let C = C(z 0 ; δ ) and = f (C), it follows by the Argument Principle (10.9) that 1 f (z) 1= dz 2πi C f (z) − α 1 1 dω dω = = 2πi ω − α 2πi ω − β

13.1 Conformal Equivalence

171

for all β in some sufficiently small disc D(α; ), since the winding number is locally constant (see following 10.3). If we then take δ ≤ δ so that D(z 0 ; δ) ⊂ f −1 (D(α; )) it follows that for any z 1 , z 2 ∈ D(z 0 ; δ)

1= =

1 2πi 1 2πi

C

1 dω = ω − f (z 1 ) 2πi

dω ω − f (z 2 ) 1 f (z) f (z) dz = dz; f (z) − f (z 1 ) 2πi C f (z) − f (z 2 )

i.e., the values f (z 1 ) and f (z 2 ) are both assumed once inside C so that f (z 1 ) = f (z 2 ) if z 1 = z 2 .

E XAMPLE 1 f (z) = e z has a nonzero derivative at all points, hence it is everywhere conformal and locally 1-1. (Note that is it not globally 1-1 since f (z + 2πi ) = f (z).) By the conformality of f , the images of the orthogonal lines x = constant and y = constant under the mapping f are themselves orthogonal. We leave it as an exercise to verify this by showing that f maps the vertical lines x = constant onto circles centered at the origin and maps the horizontal lines y = constant onto rays from the origin.

ez 0

0

♦

E XAMPLE 2 Let f (z) = z 2 . Since f (z) = 2z = 0 except at z = 0, f is conformal throughout z = 0. Thus, if we set f = u + i v, it follows that the preimages of the curves u = c1 , ν = c2 for c1 , c2 = 0 must be orthogonal. Indeed, since u(z) = x 2 − y 2 , v(z) = 2x y

172


these preimages are the orthogonal systems of hyperbolas given by

0

y

z

x

z2

0

v

ω

u

x 2 − y 2 = c1 , 2x y = c2 (see the figure below).

♦ To analyze the mapping properties of a function f at a point z where f (z) = 0, we first consider the following special case.

13.1 Conformal Equivalence

173

13.5 Definition Let k be a positive integer. f is a k-to-1 mapping of D1 onto D2 if for every α ∈ D2 , the equation f (z) = α has k roots (counting multiplicity) in D1 . 13.6 Lemma Let f (z) = z k , k a positive integer. Then f magnifies angles at 0 by a factor of k and maps the disc D(0; δ), δ > 0, onto the disc D(0; δ k ) in a k-to-1 manner. Proof Since f (r eiθ ) = r k eikθ , f maps the ray from 0 with argument θ onto the ray from 0 with argument kθ . Hence the angle at 0 between any two rays is magnified by a factor of k. To see that f (z) = α, α ∈ D(0; δ k ) has k roots in D(0; δ) recall that if α = 0 there are k distinct roots all lying on the circle |z| = |α|1/ k . If α = 0, the equation z k = α has a k-fold root at the origin. We can now “complete” Theorem 13.4. 13.7 Theorem Suppose f is analytic at z 0 with f (z 0 ) = 0. Then, unless f is constant, in some sufficiently small open set containing z 0 , f is a k-to-1 mapping and f magnifies angles at z 0 by a factor of k, where k is the least positive integer for which f (k) (z 0 ) = 0. Proof We may assume, without loss of generality, that f (z 0 ) = 0. [Otherwise, we could first consider f (z) − f (z 0 ).] Then, by hypothesis, the power series expansion of f about z 0 is of the form f (z) = ak (z − z 0 )k + ak+1 (z − z 0 )k+1 + ak+2 (z − z 0 )k+2 + · · · ' ( = (z − z 0 )k ak + ak+1 (z − z 0 ) + ak+2 (z − z 0 )2 + · · · with ak = f (k) (z 0 )/k! = 0. If we let g(z) represent the bracketed power series, we note that g(z 0 ) = 0 so that g has an analytic k-th root in some disc D(z 0 ; δ) (see the comments following Theorem 8.8). Thus, in that disc, f (z) = [h(z)]k where h is an analytic function defined by h(z) = (z − z 0 )g 1/k (z) and

h(z 0 ) = 0, h (z 0 ) = g 1/k (z 0 ) = 0.

174


Hence, in a sufficiently small neighborhood D of z 0 , f is the composition of the 1-1 and conformal mapping h followed by the mapping z k . Since z k magnifies angles at 0 by a factor of k, it follows that f magnifies angles at z 0 by k. Also, since z k is k-to-1 on discs about 0, it follows that if D(0; ) ⊂ h(D) and

η = h −1 (D(0; )),

then f is k-to-1 on η.

D

η f(D)

δ

∋

z0

f

k

0

h zk

h(D)

∋ 0

The previous results combine to yield the following properties of 1-1 analytic functions. 13.8 Theorem Suppose f is a 1-1 analytic function in a region D. Then a. f −1 exists and is analytic in f (D), b. f and f −1 are conformal in D and f (D), respectively. Proof Since f is 1-1, f = 0. Hence f −1 is also analytic (Proposition 3.5). Furthermore, ( f −1 ) = 1/ f so that f −1 also has a nonzero derivative. Thus f and f −1 are both conformal.

13.2 Special Mappings

175

Theorem 13.8 motivates the following definitions: 13.9 Definitions a. A 1-1 analytic mapping is called a conformal mapping. b. Two regions D1 and D2 are said to be conformally equivalent if there exists a conformal mapping of D1 onto D2 . We leave it as an exercise to verify that “conformal equivalence” satisfies the usual axioms of an equivalence relation. In particular, we note that the transitive property follows from the fact that the composition of two conformal mappings is also a conformal mapping, and we will use this fact throughout the remainder of the chapter. The Riemann Mapping Theorem, which we will prove in the next chapter, asserts that any two simply connected domains (besides the whole plane) are conformally equivalent. In the next section, we will consider certain special transformations that will enable us to explicitly display conformal mappings between many familiar simply connected regions.

13.2 Special Mappings I Elementary Transformations (i) ω = az + b. The linear map ω = az +b is a 1-1 analytic map of the entire plane onto itself. The effect of the mapping on a given domain can be seen by viewing it as a composition ω = ω3 ◦ ω2 ◦ ω1 of the mappings 1. ω1 = kz, k = |a| 2. ω2 = eiθ z, θ = Arg a 3. ω3 = z + b. A mapping of the form ω = kz, k > 0 is called a magnification. It sends each point onto another point along the same ray from the origin, multiplying its magnitude by a factor of k. The mapping ω = eiθ z is a counterclockwise rotation through an angle θ . Finally, ω = z + b is called a translation since it translates each point by the complex number, or vector, b. (ii) ω = z α , α > 0. As we noted in Chapter 8, the function ω = z α given by z α = eα log z is analytic in every simply connected domain that does not contain 0. If we take the branch of log z which is real on the positive axis, then z α will also map the positive axis onto itself. The point z = r eiθ is mapped onto r α eiαθ and hence ω = z α maps the wedge S = {z : θ1 < Arg z < θ2 } onto the wedge T = {ω : αθ1 < Arg ω < αθ2 }. If, moreover, αθ2 − αθ1 ≤ 2π; i.e., if θ2 − θ1 ≤ 2π/α, the mapping is a conformal mapping of S onto T .

176


Some examples are sketched below:

z2 0

π– 4

z½ 0

0

(iii) ω = e z . Since e z = e x eiy , the function ω = e z maps the strip: y1 < y < y2 onto the wedge: y1 < Arg ω < y2 . If y2 − y1 ≤ 2π, the mapping is 1-1. For example, the strip 0 < y < π is mapped conformally onto the upper half-plane.

πi

ω

z log ω

0

0 ez


177

II The Bilinear Transformation ω = (az + b)/(cz + d) The mapping given by f (z) =

az + b , ad − bc = 0 cz + d

(1)

is called a bilinear transformation. The condition ad − bc = 0 insures that f is neither identically constant nor meaningless. Since f (z) =

ad − bc = 0, (cz + d)2

f is locally 1-1 and conformal. In fact, a bilinear transformation is globally 1-1 since az 2 + b az 1 + b = cz 1 + d cz 2 + d implies that (ad − bc)(z 1 − z 2 ) = 0 and hence that z1 = z2. The bilinear transformation (1) maps the full plane, minus the point −d/c, onto the full plane minus the point a/c, since the equation az + b =ω cz + d has the explicit solution z=

a dω − b for every ω = . −cω + a c

In fact, if we consider the limiting values f (∞) = (a/c) and f (−d/c) = ∞ we can say that f is a 1-1 mapping of the Riemann sphere onto itself. (See Section 1.5.) The set of bilinear transformations forms a group under composition. It is easily seen that the inverse of a bilinear transformation is also bilinear since, as above, ω=

az + b cz + d

admits the solution z=

dω − b , and da − (−b)(−c) = ad − bc = 0. −cω + a

We leave the verification of the other group properties as an exercise. A very useful property of bilinear transformations is that they map circles and lines onto other circles and lines. We prove this first for the special case f (z) = 1/z.

178


13.10 Lemma If S is a circle or line, and f (z) = 1/z, then f (S) is also a circle or line. Proof (A proof involving the Riemann sphere is outlined in Exercises 27 and 28 of Chapter 1. The following proof is more direct.) Assume first that S = C(α; r ) and let 1 f (S) = ω = : z ∈ S . z Writing the equation for S in the form (z − α)(¯z − α) ¯ = r2 we have z z¯ − α z¯ − αz ¯ = r 2 − |α|2 or, in terms of ω, α α¯ 1 − − = r 2 − |α|2 . (1) ωω¯ ω¯ ω Note then that if r = |α|; i.e., if S passes through the origin, (1) is equivalent to 1 − αω − α¯ ω¯ = 0 or

1 . 2 In that case, if α = x 0 + i y0 and ω = u + i ν, the equation for ω becomes Re αω =

ux 0 − vy0 =

1 ; 2

i.e., f (S) is a line in the ω-plane. If, on the other hand, r = |α|, then (1) is equivalent to −1 α¯ α ωω¯ − ω¯ − ω= , 2 2 2 2 2 |α| − r |α| − r |α| − r 2 and setting β=

α¯ − r2

|α|2

we obtain ¯ + |β|2 = ωω¯ − β ω¯ − βω

r2 . (|α|2 − r 2 )2


179

Thus

|ω − β|2 =

r 2 |α| − r 2

2 ,

so that f (S) is a circle with center β and radius |r/(|α|2 − r 2 )|. Finally, if S is a straight line, then there exist real-valued a, b, c such that if z = x + i y ∈ S, ax + by = c. (2) Letting α = a − bi , (2) is equivalent to Re αz = c or αz + α¯ z¯ = 2c. It follows then, as above, that f (S) is either a circle or a line.

13.11 Theorem az + b , ad − bc = 0, cz + d maps circles and lines onto circles and lines. f (z) =

Proof If c = 0, f is a linear map and the result is immediate. Otherwise, we can write 1 acz + ad − ad + bc 1 az + b ad − bc = = a− . cz + d c cz + d c cz + d Thus f is the composition f = f3 ◦ f 2 ◦ f 1 , where f 1 (z) = cz + d, 1 f 2 (z) = , z a ad − bc z. f 3 (z) = − c c f 1 and f 3 are linear; hence they map circles and lines into circles and lines. By Lemma 13.10, the same is true for f 2 , and thus it follows that f has the desired property. The above properties of bilinear transformations make them a very handy tool in solving many conformal mapping and miscellaneous geometric problems.

180


E XAMPLE 1 Find a conformal mapping f of the semi-disc S = {z : |z| < 1, Im z > 0} onto the upper half-plane.

f –1

Note that because g(z) = 1/(z + 1) has a pole at −1, it maps the line segment [-1, 1] and the upper semi-circle onto infinite rays. Furthermore, the two rays must intersect at g(1) = 12 , and by the conformality of g, they intersect orthogonally. By considering several points, it can then be seen that g maps the segments onto the lines indicated below and maps the semi-disc onto the quadrant bounded by the orthogonal rays.

i 1 z+1 –1

1

f (1) = –¹ ²

f (0) = 1

0 f (i)

Thus, the desired mapping f is given by 1 2 −(z − 1)2 f (z) = i g(z) − = . 2 4(z + 1)2 ♦


181

E XAMPLE 2 Suppose two circles C1 and C2 are tangent at a point α and a chain of circles tangent to C1 and C2 and to each other is constructed as indicated. Prove that the points of tangency a, b, c, . . . thus created all lie on a circle. b

c

a d

α C1

C2

Consider the image of the above diagram under the mapping f (z) = 1/(z − α). Since the mapping is 1-1 and has a pole at α, C1 and C2 are mapped into a pair of parallel lines. Furthermore, all the other circles are mapped into circles and, again since f is 1-1, the circles will be tangent to the parallel lines and to each other.

C1

α

f

C2

It is clear that the points f (a), f (b), f (c), . . . lie on a straight line (between f (C1 ) and f (C2 )). Finally, then, a, b, c . . . all lie on the image of this line under the inverse transformation f −1 . Since f −1 is also bilinear, this image is a circle and the result is established. ♦

182


In light of Theorem 13.11, it will come as no surprise that bilinear functions can be used to map half-planes and discs conformally onto other half-planes and discs. In fact, as we shall see below, all such mappings are given by bilinear transformations. 13.12 Definition A conformal mapping of a region onto itself is called an automorphism of that region. 13.13 Lemma Suppose f :D1 → D2 is a conformal mapping. Then a. any other conformal mapping h:D1 → D2 is of the form g ◦ f ; b. any automorphism h of D1 is of the form f −1 ◦ g ◦ f, where g is an automorphism of D2 . Proof a. If f and h are both conformal mappings of D1 onto D2 , then h ◦ f −1 is an automorphism of D2 ; i.e., h ◦ f −1 = g and h = g ◦ f . b. If h is an automorphism of D1 , f ◦ h ◦ f −1 is an automorphism of D2 ; thus f ◦ h ◦ f −1 = g and h = f −1 ◦ g ◦ f . We now consider the problem of determining all the automorphisms of the unit disc. 13.14 Lemma The only automorphisms of the unit disc with f (0) = 0 are given by f (z) = eiθ z. Proof If f maps the unit disc 1-1 onto itself and f (0) = 0, then by Schwarz’ Lemma (7.2) | f (z)| ≤ |z| for |z| < 1. Moreover, since argument,

f −1

also maps the disc onto itself and | f −1 (z)| ≤ |z| for |z| < 1.

(3) f −1 (0)

= 0, by the same (4)

However, (3) and (4) can both be valid only if | f (z)| = |z| and, by Schwarz’ Lemma once again, it follows that f (z) = eiθ z. Suppose now that we wish to find an automorphism f of the unit disc with f (α) = 0, for a fixed α, 0 < |α| < 1. If we assume that f is bilinear, then since f is globally 1-1, it must map the unit circle onto itself and we can thus apply the Schwarz Reflection Principle (7.8) (see also Exercise 19, Chapter 7) to conclude that


183

f (1/α) ¯ = ∞. Hence f must be of the form z−α . f (z) = c z − 1/α¯ Setting | f (1)| = |cα| = 1 we have |c| = (1/|α|), and f may be written in the form z−α iθ f (z) = e . 1 − αz This suggests the following theorem. 13.15 Theorem The automorphisms of the unit disc are of the form z−α g(z) = eiθ , |α| < 1. 1 − αz ¯ Proof Let g(z) = (z−α)/(1− αz). ¯ Then, as we noted previously (following 7.2), |g(z)| = 1 for |z| = 1. Since g(α) = 0, it follows that g is indeed an automorphism of the unit disc. Now assume that f is an automorphism of the unit disc with f (α) = 0. Then h = f ◦ g −1 is an automorphism with h(0) = 0, so that according to the previous lemma h(z) = eiθ z

or f (z) = e

iθ

z−α 1 − αz ¯

.

Suppose next that we wish to determine a conformal mapping h of the upper halfplane onto the unit disc. Again, let us first assume that h is bilinear and h(α) = 0, for fixed α with Im α > 0. Then, since the real axis is mapped into the unit circle, it follows by the Schwarz Reflection Principle that h(α) ¯ = ∞ so that z−α h(z) = c . z − α¯ 13.16 Theorem The conformal mappings h of the upper half-plane onto the unit disc are of the form z−α , Im α > 0. h(z) = eiθ z − α¯

184


Proof Let f (z) = (z − α)/(z − α). ¯ Since |z − α| = |z − α| ¯ for real z, f maps the real axis onto the unit circle. Also, since f (α) = 0 and Im α > 0, it follows that f maps the upper half-plane onto the unit disc. Suppose then that h is any conformal mapping of the upper half-plane onto the unit disc and h(α) = 0. By Lemma 13.13, h is of the form h=g◦ f where g is an automorphism of the disc. However, since h(α) = g(0) = 0, it follows that g(z) = eiθ z (13.14) and z−α . h(z) = eiθ z − α¯ 13.17 Theorem The automorphisms of the upper half-plane are of the form h(z) =

az + b cz + d

with a, b, c, d real and ad − bc > 0. Proof Let h be as above. Then clearly h maps the real axis onto itself. Also, Im f (i ) =

ad − bc > 0, c2 + d 2

so that i is mapped into the upper half-plane and hence f is an automorphism of the upper half-plane. To show that there are no other automorphisms, we can apply Lemma 13.13 and Theorem 13.15 to show that any such automorphism h must be of the form h = f −1 ◦ g ◦ f with z−i z−α and g(z) = eiθ , |α| < 1. f (z) = z+i 1 − αz ¯ We leave it as an exercise to verify that such a mapping can be written in the form h(z) =

az + b ; cz + d

a, b, c, d, r eal;

ad − bc > 0.

(See Exercise 16.)

In the results that follow, we will see that there is a unique bilinear mapping sending any three distinct points z 1 , z 2 , z 3 onto any three distinct points ω1 , ω2 , ω3 , respectively. 13.18 Definition z 0 is called a fixed point of the function f if f (z 0 ) = z 0 .


185

13.19 Proposition A bilinear transfomation (other than the identity mapping f (z) = z) has at most two fixed points. Proof Let f (z) = (az + b)/(cz + d). If c = 0, the equation f (z) = z is equivalent to the quadratic equation az + b = cz 2 + dz and hence has at most two solutions. (Note also that in this case f (∞) = a/c = ∞.) If c = 0, f is linear and, unless a/d = 1, f (z) = z has one solution in the finite complex plane. In this case, since f (∞) = ∞, the point at infinity may be considered a second fixed point. Finally, if f (z) = z + b, there are no fixed points in C. 13.20 Lemma The unique bilinear mapping sending z 1 , z 2 , z 3 into ∞, 0, 1, respectively, is given by (z − z 2 )(z 3 − z 1 ) T (z) = . (z − z 1 )(z 3 − z 2 ) Proof Certainly T has the desired properties. If S is another bilinear transformation which maps z 1 , z 2 , z 3 into ∞, 0, 1, then T ◦ S −1 is a bilinear map with three fixed points, so that T ◦ S −1 is the identity map and T ≡ S. Note that the lemma, with the appropriate modifications, is valid also if some z i = ∞. If z 1 = ∞, the map is given by T (z) =

z − z2 . z3 − z2

T (z) =

z3 − z1 ; z − z1

T (z) =

z − z2 . z − z1

If z 2 = ∞, and if z 3 = ∞,

13.21 Definition The cross-ratio of the four complex numbers z 1, z 2 , z 3 , z 4 —denoted (z 1, z 2 , z 3 , z 4 )— is the image of z 4 under the bilinear map which maps z 1 , z 2 , z 3 into ∞, 0, 1, respectively. By the preceding lemma z3 − z1 z4 − z2 (z 1 , z 2 , z 3 , z 4 ) = . z4 − z1 z3 − z2

186


13.22 Proposition The cross-ratio of four points is invariant under bilinear transformations: i.e., if S is bilinear, (Sz 1 , Sz 2 , Sz 3 , Sz 4 ) = (z 1 , z 2 , z 3 , z 4 ). Proof [Ahlfors] Let T be the bilinear map which sends z 1 , z 2 , z 3 into ∞, 0, 1. Then T ◦ S −1 maps Sz 1 , Sz 2 , Sz 3 into ∞, 0, 1 and by definition (Sz 1 , Sz 2 , Sz 3 , Sz 4 ) = T ◦ S −1 (Sz 4 ) = T z 4 = (z 1 , z 2 , z 3 , z 4 ).

13.23 Theorem The unique bilinear transformation ω = f (z) mapping z 1 , z 2 , z 3 into ω1 , ω2 , ω3 , respectively, is given by (z − z 2 )(z 3 − z 1 ) (ω − ω2 )(ω3 − ω1 ) = . (ω − ω1 )(ω3 − ω2 ) (z − z 1 )(z 3 − z 2 )

(5)

Proof The existence of the mapping f is easily established. If we let T1 , T2 be the bilinear maps with T1 : z 1 , z 2 , z 3 → ∞, 0, 1 T2 : ω1 , ω2 , ω3 → ∞, 0, 1 T2−1

◦ T1 . To show that ω = f (z) must satisfy (5), we need only invoke then f = Proposition 13.22 that the cross-ratio of any four points is preserved under a bilinear map and hence (ω1 , ω2 , ω3 , ω) = (z 1 , z 2 , z 3 , z). (The appropriate modification of (5) if some z i or some ωi = ∞ is left as an exercise.) Note that (5) affords a direct method to find the desired mapping by simply solving for ω. E XAMPLE To map z 1 = 1, z 2 = 2, z 3 = 7 onto ω1 = 1, ω2 = 2, ω3 = 3 we set (z − 2)(7 − 1) (ω − 2)(3 − 1) = (ω − 1)(3 − 2) (z − 1)(7 − 2) or, solving for ω, ω=

7z − 4 . 2z + 1

♦

13.3 Schwarz-Christoffel Transformations

187

13.3 Schwarz-Christoffel Transformations I

Mapping a Semi-Infinite Strip Onto a Half-Plane We will show that f (z) = sin z maps the semi-infinite strip: π −π < Re z < ; Im z > 0 2 2

conformally onto the upper half-plane by considering its behavior on the rectangle R: −π π ≤ Re z ≤ ; 0 ≤ Im z ≤ N 2 2 for large N. ) * π Of course, the interval −π 2 , 2 is mapped onto [−1, 1]. For complex z, we use the identity sin z = sin(x + i y) = sin x cosh y + i cos x sinh y

(1)

sin( π2

to show that + i y) = cosh y, which is real-valued and increases from 1 to cosh N as y increases from 0 to N. Note that the mapping f (z) = sin z doubles the vertex angle of R at z = π2 , as we could have anticipated since f (z) has a simple zero at that point. Along the line Im z = N, sin z = sin x cosh N + i cos x sinh N

(2)

For large N, sinh N is just slightly smaller than cosh N since both are very close to 12 e N . Hence, according to (2), as x varies from π2 to −π 2 , sin z traces an "almost circular" elliptical path from cosh N counterclockwise to − cosh N. −π Finally, sin z maps the interval connecting −π 2 + i N to 2 onto the interval [ − cosh N, −1]. So sin z maps the boundary of R onto the boundary of a region S whose base is the real interval [ − cosh N, cosh N] and which is very close to a semicircular region in the upper half-plane. By the Argument Principle, then, f maps the interior of R onto the interior of S (see Remark 2 following Corollary 10.9).

w = sin z

y

v

Ni

–π/2

0

π/2

x

–1

0

1

v2 u2 + ≤1 2 cosh N sinh2N

u

188


It follows, letting N → ∞, that sin z maps the semi-infinite strip: π −π < Re z < ; Im z > 0 2 2

conformally onto the upper half-plane.

A mapping of any other semi-infinite strip onto a half-plane or onto any of the domains previously considered can be found by composing sin z with the appropriate conformal mappings. It is interesting to examine the inverse function, sin−1 z, which can be defined by the familiar integral formula: z 1 −1 sin z = dζ 1 − ζ2 0

Unlike sin z, which is an entire function, sin−1 z is not analytic at the points z = ±1. This follows immediately from the fact that its derivative, √ 1 2 , approaches 1−z ∞ as z → ±1. It is also evident geometrically since sin−1 z maps the straight angles at z = ±1 onto right angles. The function sin−1 z is analytic, however, as is √ 1 2 , 1−z

in the plane slit along the two rays z = ±1 − i y, 0 ≤ y < ∞. In particular, it is analytic in the upper half-plane and continuous to the boundary, including the points 1 ±1, since the improper definite integral 0 √ 1 2 d x converges to π/2. As was the 1−x case with sin z, it is easy to determine the behavior of sin−1 z along the boundary; i.e., along the x − axi s. To that end, note that its derivative: d 1 sin−1 z = √ dz 1 − z2 −1 is positive on the interval −1 < )z < 1. So [−1, 1] * sin z maps the closed interval π π monotonically onto the interval − 2 , 2 . To see how the analytic √ 1 2 behaves 1−z √ on the remainder of the line, consider 1 + z as z varies along a small semicircular arc from −1 + r to −1 − r . That is, let z = −1 + r eiθ , 0 ≤ θ ≤ π. Then √ √ 1 + z = r eiθ = r eiθ/2 √ and at z = −1 − r , i.e. when θ = π, 1 + z is a multiple of i . It follows that √ 1 and √ 1 are negative multiples of i throughout the interval (−∞, −1). This 1+z 1−z 2 −1 property of its derivative, and the fact that −∞ √ 1 2 dζ diverges, show that sin−1 z 1−ζ √ maps (−∞, −1) onto the ray from −π/2 + i ∞ to −π/2. By analyzing 1 − z in a semicircular arc around z = 1, we can see that sin−1 z maps the interval (1, ∞) onto the ray from π/2 to π/2 + i ∞. (This also follows from the Schwarz Reflection


Principle (7.9 ) since √ 1

1−z 2

189

is real-valued on the imaginary axis, so that sin−1 z

maps the imaginary axis into itself.) While our insights into the mapping properties of sin z were very dependent on formula (1), our analysis of the behavior of sin−1 z along the real line can easily be adapted to a wide range of problems. This will ultimately lead to the general Schwarz-Christoffel formula, but first we consider the following special case: II

Mapping the Upper Half-Plane Onto a Rectangle. As we saw in the last section, if the argument of f (z) is constant along a straight line, f (z) will map that line into another line. To be specific, recall that z f (z) − f (z 0 ) =

f (ζ )dζ

z0

Hence if γ represents the ray: z = z 0 + r eiα , r > 0, and if Arg( f ) = β, then = f (γ ) will travel along the ray ω = f (z 0 ) + sei(α+β) , s > 0 More specifically, if z travels to the right along the real axis from a real point z 0 , and if f has a constant argument of θ, then f (z) will travel along the ray from f (z 0 ) with argument θ. If we want to find a function f which maps the upper half-plane onto a rectangle, we would like f to map the real line onto the four sides of the rectangle. This suggests that f should have exactly four different arguments on the segments of the real line and that its argument should increase by π2 as we move (to the right) from one segment to the next. To create such a function f , note that if z 0 represents any real number, z − z 0 has a constant argument of π for real z < z 0 , and a constant argument of 0 for real z > z 0 . If we define the analytic function (z − z 0 )−α so that it is positive for real z > z 0 , its argument will increase from −απ to 0 as z crosses the point z 0 . In particular, with α = 1/2, the argument of (z − z 0 )−α increases by π/2 as z crosses over the point z 0 . So, to define f , we can pick four arbitrary real numbers a < b < c < d and let f (z) = 1/ (z − a)(z − b)(z − c)(z − d). The square root in the denominator is defined as the product of the square roots of each of its linear factors, and each of these is defined to be positive for large positive values of z. We can then define z f (z) =

z

f (ζ )dζ = 0

0

1 dζ √ (ζ − a)(ζ − b)(ζ − c)(ζ − d)

(3)

Note that although f (z) is undefined at the (real) points a, b, c, d, it is analytic in the entire complex plane slit along the four rays from t to t − i ∞, for

190


t = a, b, c, d. So it is analytic on the closed upper half-plane minus the points a, b, c, d. Technically, the path of integration in (3) should be indented slightly to avoid these points. However, since the associated improper integrals are all convergent, the path of integration can be along the real line for all real z, including z = a, b, c, d, and f is continuous at all points of the real line. Moreover, since the real integral ∞ 1 dx √ (x − a)(x − b)(x − c)(x − d) d

converges, limz→∞ f (z) exists, as does limz→−∞ f (z). According to our earlier remarks, f maps the interval (−∞, a] onto a finite interval parallel to the real line, and it maps the four successive intervals: [a, b],[b, c], [c, d], [d, ∞) onto intervals, each of which represents a counterclockwise rotation of π/2 from its predecessor. These facts alone do not guarantee that the image of the real line is the boundary of a rectangle. It does follow, however, once we can show that f (−∞) = f (∞); i.e., that ∞ −∞

1 dx = 0 √ (x − a)(x − b)(x − c)(x − d)

(4)

This follows by the type of argument used in Chapter 11. Let C R be the closed contour consisting of the real interval [−R, R] followed by R , the upper semicircle of radius R, traversed counterclockwise from R to −R, and let C R,ε be the contour formed by replacing each interval in C R of the form [t − ε, t + ε] , t = a, b, c, d with a semicircle in the upper half-plane centered at t, with radius ε. By the Cauchy Closed Curve Theorem, f (ζ )dζ = 0 C R,ε

and, letting ε → 0, we see that

f (ζ )dζ = 0

CR

That is,

R −R

1 dx + √ (x − a)(x − b)(x − c)(x − d)

f (ζ )dζ = 0

R

Since | f (z)| is asymptotic to 1/R 2 throughout R , the usual M − L estimate shows that the second integral above approaches 0 as R → ∞, thus proving equation (4). An analogous argument can be used to show that f maps the upper half-plane conformally onto the inside of the rectangle. We need only note that for any point ω inside the rectangle, if R is sufficiently large, f maps C R onto a contour which is


191

just a slight perturbation of the boundary of the rectangle and hence winds around the point ω exactly once. By the Argument Principle, f takes the value ω exactly once inside C R . The general result follows by letting R → ∞. It is interesting that we never directly established that the definite integrals which yield the lengths of opposite sides of the rectangle have the same magnitude. In fact, if it weren’t an obvious corollary of our other arguments, it would be hard to verify directly. For example, if we let a, b, c, d equal 1, 2, 5, 9, respectively, it follows that 2 1

1 dx = √ (x − 1)(2 − x)(5 − x)(9 − x)

9 5

1 dx √ (x − 1)(x − 2)(x − 5)(9 − x)

One particularly nice choice for a, b, c, d is the set of values −1/k, −1, 1, 1/k with k < 1. The resulting formula for f (z) given by (3), and with a suitable additional constant factor, is z 1 f (z) = dζ 2 (1 − ζ )(1 − k 2 ζ 2 ) 0

This is known as an elliptic integral of the first kind. In this form it is easy to verify directly that opposite sides of the rectangle obtained have equal length. It is also easy to show that by choosing an appropriate value of k, the rectangle obtained can have adjacent sides of any desired ratio. Note also that if we omitted the fourth point d in formula (3), the resulting function z f (z) =

z

f (ζ )dζ = 0

0

1 dζ √ (ζ − a)(ζ − b)(ζ − c)

would still map the (closed) upper half-plane, with the point at ∞, onto a closed rectangle. This follows from the behvior of f (z) along the real axis and from the fact that in this case, as in formula (3), f (−∞) = f (∞). Here, the point at infinity takes the place of the "missing" point d, and is mapped by f onto one of the vertices of the rectangle. III

Mapping the Upper Half Plane Onto any Convex Polygon The ideas of the previous section are easily generalized to find a conformal mapping f of the upper half-plane onto a convex polygon with any number of sides and any interior angles. To assure that f maps the real line (with the point at infinity) onto the boundary of such a polygon, we choose n real points a1 < a2 < · · · < an , and define f (z) = (z − a1 )−α1 (z − a2 )−α2 · · · (z − an )−αn where the n exterior angles of the polygon are equal in order to α1 π, α2 π, . . . , αn π. As in the previous section, each of the analytic functions (z −ai )−αi is defined so that its argument is 0 for real z > ai and −αi π for real z < ai . Since the desired image

192


polygonis convex, 0 < αi < 1 for each i , and αi = 2. So if we once again define z f (z) = 0 f (ζ )dζ , it follows that the argument of f increases by αi π, and f has a corresponding change in direction, as z crosses the real point ai . Moreover, by the same reasoning used in the previous section, limz→∞ f (z) and limz→−∞ f (z) both exist and are equal. Taken together, these properties assure that f is a conformal mapping of the upper half-plane onto a polygon of the desired type. Here, too, if we omitted the final factor: (z − an )αn from the formula for f , it would still map the upper half-plane onto a polygon of the desired type. As in the case of the rectangle, the point at ∞ would map into one of the n vertices of the polygon. In this general setting, it is more difficult to show that, with the proper choices of a1 , a2 , . . . an , the function f can be made to map the upper half-plane onto an arbitrary polygon; that is, onto a polygon with any desired shape. On the other hand, it is not hard to define the mappings onto certain special polygons. This is especially easy for triangles since their shape is entirely determined by their angles. Thus, it follows e.g. that z f (z) =

(ζ 2 − 1)−2/3 dζ

0

maps the upper half-plane onto an equilateral triangle.

Note: The techniques used above can be applied equally well to finding mappings from the upper half-plane onto a wide variety of polygonal regions. Many such examples, and much more information regarding Schwarz-Christoffel mappings, can be found in the classic book of Nehari.

Exercises 1. Verify directly that f (z) = z k is locally 1-1 for z = 0, k any nonzero integer. 2. Find the image under ω = ez of the lines x = constant and y = constant. 3. Find a conformal mapping f between the regions S and T , where i. ii. iii. iv.

S S S S

= {z = x + iy : − 2 < x < 1}; T = D(0; 1) = T = the upper half-plane; f (−2) = −1, f (0) = 0 and f (2) = 2 = {reiθ : r > 0 and 0 < θ < π/4}; T = {x + iy : 0 < y < 1} = D(0; 1)\[0, 1]; T = D(0; 1).

[Hint: For (iv) use the mapping of the upper semi-disc onto a quadrant.] 4.* Find a conformal mapping of the region "between" the circles: |z| = 2 and |z − 1| = 1 onto the unit disc. 5.* Find a conformal mapping of the semi-infinte strip: x > 0, 0 < y < 1 onto the unit disc. 6.* Find a conformal mapping of the semi-disc S = {z : |z| < 1, Im z > 0} onto the unit disc.

Exercises

193

7.* Verify that “conformal equivalence” satisfies the reflexive, symmetric and transitive properties of an equivalence relation. 8. a. Prove that a linear function maps polygons onto polygons. b. Suppose f is entire and, for some rectangle R, f (R) is a rectangle. Prove f is linear. 9. Prove that bilinear mappings form a group under composition. 10. Find the image of the circle |z| = 1 under the mappings 1 , z 1 , b. ω = z−1 1 . c. ω = z−2 a. ω =

11. Show that the only automorphism of the unit disc with f (0) = 0, f (0) > 0 is the identity map f (z) ≡ z. 12. Suppose f 1 and f 2 are both conformal mappings of a region D onto the unit disc and for some z 0 ∈ D, f 1 (z 0 ) = f 2 (z 0 ) = 0;

f 1 (z 0 ), f 2 (z 0 ) > 0.

Prove f 1 ≡ f 2 . 13. Show that all conformal mappings of a half-plane or disc onto a half-plane or disc are given by bilinear transformations. 14. What is the image of the upper half-plane under a mapping of the form f (z) =

az + b a, b, c, d real; cz + d

ad − bc < 0?

15. Find a formula for all the automorphisms of the first quadrant. 16. Complete Theorem 13.17 by showing h is of the form h(z) =

az + b a, b, c, d real; cz + d

ad − bc > 0.

[Hint: Write h = h 1 ◦ h 2 where z − i −1 iθ z−i ◦e z◦ z+i z+i z−i z − i −1 z−α ◦ . h 2 (z) = ◦ z+i 1 − αz ¯ z+i

h 1 (z) =

Show, then that h 1 (z) =

(1 + cos θ )z + sin θ (− sin θ )z + (1 + cos θ )

h 2 (z) =

(1 − Re α)z + Im α .] (Im α)z + (1 + Re α)

17. Find the fixed points of the mappings z z−1 , b. ω = . a. ω = z+1 z+1 18. Prove that (z 1 , z 2 , z 3 , z 4 ) is real-valued if and only if the four points z 1 , z 2 , z 3 , z 4 lie on a circle or line.

194


19. Find the bilinear mappings which send a. 1, i, −1 onto −1, i, 1, respectively b. −i, 0, i onto 0, i, 2i, respectively c. −i, i, 2i onto ∞, 0, 13 respectively. 20. Find a conformal map f of the region between the two circles |z| = 1 and |z − 14 | = 14 onto an annulus a < |z| < 1. [Hint: Find a bilinear map which simultaneously maps |z| < 1 onto |z| < 1 and |z − 14 | < 14 onto a disc of the form |z| < a.] z 21. * Find the image of the upper half-plane under the mapping f (z) = function related to sin−1 z ?

1/ ζ 2 − 1dζ . How is this

0

22. * Find a mapping of the upper half-plane onto an isosceles right triangle. 23. * Find a mapping of the upper half-plane onto a square. [Hint: Let the point at infinity map onto one of the vertices of the square.]

Chapter 14

The Riemann Mapping Theorem

14.1 Conformal Mapping and Hydrodynamics Before proving the Riemann Mapping Theorem, we examine the relation between conformal mapping and the theory of fluid flow. Our main goal is to motivate some of the results of the next section and the treatment here will be less formal than that of the remainder of the book. Consider a fluid flow which is independent of time and parallel to a given plane, which we take to be the complex plane. The flow (or velocity) function g is then a two-dimensional or complex variable of two variables and we can write it in the form g(z) = u(z) + i v(z) where u and v are real-valued. If we let σ and τ denote, respectively, the circulation around and the flux across a closed curve C, it can be shown that g(z)dz = σ + i τ (see Appendix II) . (1) C

We will confine our attention to incompressible fluids and flows which are locally irrotational and source-free. That is, for any point z in our domain D, we assume there exists a δ > 0 such that the circulation around and flux across any closed curve C in D(z; δ)is zero. Thus, for all such curves, if we define f (z) = g(z) it follows by (1) that C f (z)dz = 0. We will assume moreover that g (and hence f ) is continuous so that, by Morera’s Theorem (7.4), f is analytic. Conversely, given an analytic f = u − i v in a domain D, its conjugate g = u + i v can be viewed as a locally irrotational and source-free flow in D.

195

196

14 The Riemann Mapping Theorem

E XAMPLES i. Suppose f (z) = k. Then g(z) = k¯ represents a constant flow throughout C.

The velocity vectors – g(z) = k

ii Let f (z) = z. Then g(z) = z¯ represents a flow which is tangent to the real and imaginary axes.

g(z) = z–

iii. Let f (z) = 1/z, z = 0. Then g(z) = f (z) =

x + iy x 2 + y2

denotes a flow whose direction at z is the same as that of the vector (from 0) to z. In this case f (z)dz = 2πi, |z|=δ

14.1 Conformal Mapping and Hydrodynamics

197

so that there is a nonzero flux across a circle centered at 0. Nevertheless the flow is locally irrotational and source-free in z = 0. (The flow is said to have a “source” at the origin.) ♦

g(z) = –1– z

As the examples above suggest, the possible fluid flows of the type considered are as abundant as the analytic functions in a given region. To focus on the particular flow related to a canonical conformal mapping of a region D, we make the following further assumptions. Al. D˜ is the closure of a bounded simply-connected region. (We will refer to D˜ as a “barrier.”) A2. g(z) = 1 at ∞. That is, limz→∞ g(z) = 1. A3. g has the direction of the tangent at the boundary of D (except for isolated points at which it may be zero or infinite). A4. The flow is totally irrotational and source-free; i.e., C f (z)dz = 0 for every closed curve C contained in D. z Under the above assumptions, suppose z 0 ∈ D and set F(z) = z0 f (ζ )dζ . By assumption (A4), F is well-defined and hence analytic in D. Moreover, according to (A2), F(z) ∼ z at ∞. Finally, suppose the boundary of D is given by z(t), a ≤ t ≤ b. Then d F(z(t)) = F (z(t))˙z (t) = g(z(t))˙z (t), dt which is real-valued according to (A3). Hence F maps ∂ D onto a horizontal segment. The converse is equally valid. If F maps D conformally onto the exterior of a horizontal interval and F(z) ∼ z at ∞, then g(z) = F (z) will represent a fluid flow in D, satisfying assumptions (A1)–(A4). E XAMPLES i. F(z) = z + 1/z maps the exterior of the unit disc conformally onto the exterior of the interval [ − 2, 2], and clearly F(z) ∼ z at ∞. Thus g(z) = F (z) = 1 −

1 z2

198


represents a fluid flow in the given region, satisfying (A1)–(A4). Note that g(−1) = g(1) = 0 and that the maximum speed is assumed at ±i where |g(z)| = 2.

0

1

ii. Suppose D is the complement of the interval I from −i to i . Then an √ √ analytic 1 + z 2 can be defined there (see Chapter 10, Exercise 16). Note that if 1 + z 2 is taken to be positive on the positive axis, √it is negative on the negative axis and √ maps D onto the exterior of [ − 1, 1]. Also 1 + z 2 ∼ z at ∞ so that F(z) = 1 + z 2 is the desired conformal mapping and the flow is given by g(z) =

z

√ 1 + z2

.

In this (idealized) case, g(0) = 0 and g(±i ) = ∞.

♦

i

–i

The examples above show how the appropriate mapping function enables us to obtain the fluid flow in a region. On the other hand, certain physical properties of the flow yield the following insights into conformal mapping. I Existence and Uniqueness of Conformal Mappings As we have seen, the existence of a conformal mapping of a “single-barrier” domain onto the exterior of a

14.1 Conformal Mapping and Hydrodynamics

199

horizontal interval is equivalent to the existence of a flow function satisfying (A1)– (A4). However, it is known that a flow satisfying these assumptions exists and is unique. According to Kelvin’s Theorem [Milne-Thomson, p.95], the totally irrotational flow of a fluid occupying a region of the type considered (with the conditions at ∞ and along the boundary) is the unique flow with the least possible kinetic energy. The proof that such a unique flow exists can thus be given in terms of the partial differential equations governing the flow. We will not proceed that far with the physics, but we will be guided by the notion that a conformal mapping of the type sought is given by the solution to an extremal problem. We cast the problem in mathematical terms and complete the details in the next section. II Conformal Mapping of Other Types of Domains By considering fluid flow throughout other types of domains, we can identify canonical domains to which they can be conformally mapped. In fact, reasoning like the above suggests that all domains with n “barriers” can be conformally mapped onto the plane slit along n horizontal line segments. In the case of a simply connected domain D1 , the standard canonical domain is the unit disc. For, if we fix z 0 ∈ D1 , the mapping given by F1 (z) = 1/(z − z 0 ) maps D1 onto a single barrier domain D2 , sending z 0 into ∞. We then can map D2 conformally onto D3 , the exterior of a horizontal interval, by a mapping F2 . Similarly, the unit disc U is mapped

z0 F1 D1

F2

D2

ϕ D3 D4 G2 0

G1

U

1

U

ϕ = G1–1 ° G2–1 ° F2 ° F1

by G 1 (z) = 1/z onto the exterior of the unit circle, D4 . Because D4 is a single barrier domain, we have a conformal mapping G 2 of D4 onto D3 . Finally, the mapping −1 ϕ = G −1 1 ◦ G 2 ◦ F2 ◦ F1

200


maps the simply-connected domain D1 conformally onto the unit disc U . Note that in changing our canonical region from the exterior of an interval to the unit disc the condition F(∞) = ∞ is replaced by ϕ(z 0 ) = 0.

14.2 The Riemann Mapping Theorem The Riemann Mapping Theorem, in its most common form, asserts that any two simply connected, proper subdomains of the plane are conformally equivalent. That is, if R1 , R2 (= C) are simply connected regions, there exists a 1-1 analytic map of R1 onto R2 . Note that the condition that R1 , R2 = C is necessary, as a consequence of Liouville’s Theorem. See Exercise 10. To prove the theorem, it suffices to show that for any simply connected region R(= C) there exists a conformal mapping of R onto U . For then, if R1 , R2 are two simply connected, proper subdomains of C, we have conformal mappings f 1 :R1 → U f 2 :R2 → U and g = f 2−1 ◦ f 1 is a conformal mapping of R1 onto R2 .

R1

f1

f2

U

R2

g = f2–1 ° f1

It is an easy exercise to show that given a conformal mapping f of R onto U and z 0 ∈ R, one can compose f with an appropriate automorphism of U so that the composite function ϕ maps R conformally onto U with the added properties that ϕ(z 0 ) = 0 and ϕ (z 0 ) > 0. [See Exercise 6.] In fact, if we insist that ϕ be a conformal mapping of R onto U with these two additional properties, then the mapping is unique. We first prove the uniqueness and then we will prove the Riemann Mapping Theorem by showing that such a unique mapping ϕ exists. Riemann Mapping Theorem For any simply connected domain R(= C) and z 0 ∈ R, there exists a unique conformal mapping ϕ of R onto U such that ϕ(z 0 ) = 0 and ϕ (z 0 ) > 0.

14.2 The Riemann Mapping Theorem

201

Proof (Uniqueness) Suppose ϕ1 and ϕ2 were two mappings with the above properties. Then = ϕ1 ◦ϕ2−1 would be an automorphism of the unit disc with (0) = 0 and (0) > 0. By 13.14, then, (z) = eiθ z and since (0) = eiθ > 0, it follows that is the identity mapping. Hence, ϕ1 ≡ ϕ2 . (Existence) As we mentioned in the last section, we will find ϕ as the solution to an extremal problem. We recall some of the solutions to extremal problems for analytic mappings of U onto U that we obtained in Chapter 7. We found that, for fixed α ∈ U , the 1-1 analytic mappings ϕ which maximize |ϕ (α)| are precisely those of the form ϕ(z) = eiθ

z−α ; 1 − αz

that is, those ϕ which i. map α onto 0 and ii. map U onto U . (See Example 2 after 7.2 and Exercises 10 and 11 of Chapter 7.) This suggests a strategy for proving the existence of the conformal mapping ϕ of an arbitrary simply connected domain R(= C) onto U . Namely, given R and z 0 ∈ R, we will consider the collection F of all 1-1 analytic functions f :R → U satisfying f (z 0 ) > 0 and take ϕ to be such that ϕ (z 0 ) = sup f ∈F f (z 0 ). The details which we must show are the following. A. F is nonempty. B. Sup f ∈F f (z 0 ) = M < ∞ and there exists a function ϕ ∈ F such that ϕ (z 0 ) = M. C. With ϕ as in (B), ϕ is a conformal mapping of R onto U such that ϕ(z 0 ) = 0 and ϕ (z 0 ) > 0. [The facts that ϕ(z 0 ) = 0 and that ϕ is an onto mapping are not guaranteed in (B).] ˜ [If R˜ contains a disc Proof of (A): Since R = C, there exists a point ρ0 ∈ R. D(ρ0 ; δ), we can simply set f (z) = δ/(z − ρ0 ) and it would clearly follow that | f | < 1 throughout R. It is possible, however, that R˜ contains no discs at all so we must use a different approach.] Since R is simply connected, there exists an analytic function 1 z − ρ0 g(z) = z 0 − ρ0 with g(z 0 ) = 1. It follows then that g must remain bounded away from −1. For if ξn − ρ0 → −1 g(ξn ) = z 0 − ρ0 then

ξn − ρ0 →1 z 0 − ρ0

202


so that ξn → z 0 . But then, by the continuity of g at z 0 , it would follow that g(ξn ) → +1 and the contradiction is apparent. Hence, for some η > 0, |g(z)+1| > η throughout R. Thus, if we set f (z) = η/(g(z) + 1), we will have | f | < 1. Since f is the composition of 1-1 functions, it too is 1-1 in R. Finally, since all the above properties are invariant under multiplication by eiθ we can assume that f (z 0 ) > 0 and hence f ∈ F . Proof of (B): Note, first, that since R is open, there exists some disc D(z 0; 2δ) ⊂ R and hence, for any f ∈ F, 1 1 f (z) | f (z 0 )| = dz ≤ , 2 2πi C(z0 ;δ) (z − z 0 ) δ by the usual M − L estimate. Suppose then that M = sup f ∈F f (z 0 ) and let f 1 , f2 , . . ., be chosen so that f n (z 0 ) → M as n → ∞. To show that there exists a function ϕ ∈ F, such that ϕ (z 0 ) = M, we will find a subsequence of { f n }∞ n=1 which converges uniformly on compacta of R. To that end, let ξ1 , ξ2 , . . . be a countable dense subset of R. (For example, the ξ s may be chosen as the points of R with rational coordinates.) Since ∞ { f n (ξ1 )}∞ n=1 is a bounded sequence, there exists a subsequence { f 1n }n=1 such that ∞ { f 1n (ξ1 )}n=1 converges to some limit which we will denote ϕ(ξ1 ). Similarly { f 1n } has a subsequence { f 2n } such that {f 2n (ξ2 )} converges and we denote its limit by ϕ(ξ2 ). Continuing in this manner, we obtain a nested sequence of subsequences ∞ ∞ {{ f kn }∞ n=1 }k=1 such that { f kn }n=1 converges at ξ1 , ξ2 . . . , ξk . If we then take the “diagonal” subsequence {ϕn (z)}∞ n=1 with ϕn = f nn , it follows that ϕn (z) converges to the function denoted by ϕ for z = ξ1 , ξ2 , . . .. We next wish to show that {ϕn } converges throughout R and uniformly on any compact subset K ⊂ R. We leave it as an exercise to show that any compact K ⊂ R is contained in a finite union of closed discs contained in R. Hence, we may assume, ˜ without loss of generality, that K is itself a fixed compact disc in R. Note that d(K, R), ˜ ˜ the distance from K to the closed set R, is positive and we can set d(K, R) = 2d > 0. Hence, since |ϕn | ≤ 1 1 1 2πd 1 ϕn (ξ ) · 2 = , z∈K |ϕn (z)| = dξ ≤ 2 2πi C(z;d) (ξ − z) 2π d d and

|ϕn (z 1 ) − ϕn (z 2 )| =

z2 z1

ϕn (z)dz

≤

|z 1 − z 2 | . d

Thus, {ϕn } is an “equicontinuous” sequence of functions on K. That is, for each > 0 and all n, |ϕn (z 1 ) − ϕn (z 2 )| ≤ as long as |z 1 − z 2 | ≤ d. If we then take z ∈ K and > 0, we can write |ϕn (z) − ϕm (z)| ≤ |ϕn (z) − ϕn (ξk )| + |ϕn (ξk ) − ϕm (ξk )| + |ϕm (ξk ) − ϕm (z)| and choosing ξk such that |ξk − z| < d/3, it follows that |ϕn (z) − ϕm (z)| <

14.2 The Riemann Mapping Theorem

203

once n and m are chosen large enough so that |ϕn (ξk ) − ϕm (ξk )|
0 is given and set S j = {z ∈ K:|ϕn (z) − ϕ(z)| < Clearly K ⊂

for n > j }.

2∞

S j . Hence, since the sets S j are open (by the equicontinuity of 2 the functions ϕn ) and K is compact, we can choose N such that K ⊂ Nj =1 S j . Thus, for all z ∈ K, |ϕn (z) − ϕ(z)| < when n > N and the convergence is uniform. Since ϕn → ϕ uniformly on compacta, ϕ is analytic (Theorem 7.6). Also, according to Theorem 10.12 ϕ (z 0 ) = lim ϕn (z 0 ) = M > 0 j =1

n→∞

so that ϕ is nonconstant. Since it is the uniform limit of 1-1 functions, ϕ is 1-1 in R (Theorem 10.15). Proof of (C): It remains only to show that ϕ(z 0 ) = 0 and that ϕ maps R onto U . To see the former, assume that ϕ(z 0 ) = α, 0 < |α| < 1. Then f (z) =

ϕ(z) − α 1 − αϕ(z)

is also a 1-1 analytic map of R into U with f (z 0 ) =

ϕ (z 0 ) . 1 − |α|2

Thus f (z 0 ) > ϕ (z 0 ), which is impossible. Assume next that ϕ(z) = ω, ω = −t 2 eiθ , 0 < t < 1. If we set g(z) = e−iθ ϕ(z), g too will map R into U, g(z 0 ) = 0 and |g (z 0 )| = ϕ (z 0 ). Moreover, g(z) = −t 2 for all z ∈ R. If we then set g(z) + t 2 f 1 (z) = 1 + t 2 g(z) it follows that f 1 maps R into U with f 1 (z 0 ) = t 2 . Since g(z) = −t 2 , f1 (z) = 0 and there exists an analytic square root f 2 (z) = f 1 (z)

204


with f2 (z 0 ) = t. Next, let f3 (z) =

f 2 (z) − t . 1 − t f 2 (z)

Clearly f 3 is 1-1 and direct calculation shows that f 1 (z 0 ) = g (z 0 )(1 − t 4 )

f (z 0 ) f (z 0 ) = 1 f 2 (z 0 ) = √1 2t 2 f1 (z 0 ) f (z 0 ) f 3 (z 0 ) = 2 2 1−t

so that, combining the above equations, f 3 (z 0 ) =

g (z 0 )(1 + t 2 ) g (z 0 ) 2t

since 1 + t 2 > 2t for 0 < t < 1. If we set f (z) = eiθ f 3 (z) we will have f ∈ F and such that f (z 0 ) > ϕ (z 0 ) which is impossible. Hence, ϕ must be onto and the proof is complete. Note: Consider the original sequence f 1 , f2 , . . . such that f n (z 0 ) → M as n → ∞. While the Riemann mapping function ϕ was obtained as the limit of a subsequence of { f n }, it turns out that the original (full) sequence { f n } converges to ϕ. For suppose there existed some subsequence f n1 , f n2 , · · · such that | f ni (z) − ϕ(z)| >

(1)

for a fixed z ∈ R and > 0. Then since f n k (z 0 ) → M as k → ∞, we could apply the previous proof to show that it has a subsequence which converges to the unique mapping function ϕ. But then (1) is impossible.

14.3 Mapping Properties of Analytic Functions on Closed Domains Introduction While the Riemann Mapping Theorem showed that any two open simply connected sets, other than C, are conformally equivalent, there is no such theorem for closed or even for compact connected sets. In fact, there is often no analytic mapping of one closed domain onto another. As an example, there is no analytic mapping of the closed upper half-plane onto the closed first quadrant. If f were such an analytic function, it would map some real number x 0 onto the origin. Suppose f (x 0 ) = 0. Then, according to Theorem 13.4, f would map the rays I1 = {x 0 + t, 0 ≤ t < ∞} and I2 = {x 0 − t, 0 ≤ t < ∞} onto two curves whose tangent lines form a straight angle.

14.3 Mapping Properties of Analytic Functions onClosed Domains

205

But two such curves, which meet at the origin, cannot both lie in the first quadrant. Similarly, if f has a zero of order k ≥ 2 at x 0 , we can complete the argument by considering the effect of f on I1 and on the ray {x 0 + teπi/k , 0 ≤ t < ∞}. The above argument might leave the impression that if we somehow “rounded the corner” of the first quadrant, the resulting region might possibly be the image of the closed upper half-plane under some analytic mapping. That, in fact, is not the case (Corollary 14.6), but it requires a separate argument. On the positive side, while an analytic mapping of one closed domain onto another is not always possible, the theorem below shows that for certain types of regions, any conformal mapping between the interiors extends to a 1-1 continuous map between the closures. Recall that a Jordan curve is a simple closed curve. As we noted in Chapter 10, the very intuitive (but difficult to prove) Jordan Curve Theorem asserts that any Jordan curve disconnects its complement in the complex plane into two disjoint regions: a bounded component known as its interior and an unbounded component known as its exterior. 14.2 Definition A region R will be called a Jordan region if it is the interior of a Jordan curve. 14.3 Theorem (Carathéodory-Osgood): Any conformal mapping between two Jordan regions can be extended to a homeomorphism between the closures of the two regions. Like the Jordan Curve Theorem, the Carathéodory-Osgood Theorem is not easy to prove. Although we will use the result throughout this section, we refer the interested reader to the proofs in the classic texts of Ahlfors and Carathéodory. A Jordan curve γ is positively-oriented if its has winding number one around its interior points. For circles, as we have seen, this is the counterclockwise direction. As with circles, the positive orientation is the one which has the interior points on the left as the curve is traversed. We can also define a triple {a1 , a2 , a3 } ⊂ ∂ R to be positively oriented with respect to ∂ R if the parametrization of ∂ R which passes through a1 , a2 , a3 in that order is positively-oriented. Suppose f is a conformal mapping between two Jordan regions R1 and R2 . Then, according to the Argument Principle (see the comments following Corollary 10.9), the induced mapping between the boundaries must be orientation-preserving. Based on the Carathéodory-Osgood Theorem, there are two additional ways to characterize a unique conformal mapping between any two Jordan regions. 14.4 Proposition: Let R be any Jordan region, and let D = D(0; 1), S = ∂D = C(0; 1). Then (i ) given a positively-oriented triple, {a1 , a2 , a3 } ⊂ ∂ R and a positively-oriented triple, {b1 , b2 , b3 } ⊂ S, there exists a unique conformal mapping f :R → D such that f (ak ) = bk for k = 1, 2, 3; (ii ) given z 0 ∈ R, a ∈ ∂ R and b ∈ S, there exists a unique conformal mapping from R to D with f (z 0 ) = 0 and f (a) = b.

206


Proof To prove (i), take any conformal mapping of R onto D and follow it with an automorphism of D which maps the images of the three boundary points of R onto the three points of S. [Such a mapping exists since Theorem 13.23 showed that there is a bilinear mapping T , sending the images of a1 , a2 , a3 onto b1 , b2 , b3 , respectively. Moreover, since both triples are positively oriented, T will also map D onto itself.] To show that the mapping is unique, use the fact that if there were two such mappings f 1 and f 2 , f 2 ◦ f 1−1 would be an automorphism of the unit disc with three fixed points on the unit circle, and hence would be the identity (Proposition 13.19). To prove (ii), follow the usual Riemann mapping with the appropriate rotation so that the image of the boundary point a is mapped onto b. The uniqueness follows from Schwarz’ Lemma. Next we would like to consider the possibility of finding analytic (not necessarily conformal or even locally conformal) mappings from one closed region onto another. It is worth recalling that in this general context the boundary of a region is not necessarily mapped entirely onto the boundary of its image. For a simple example, consider the image of the rectangle 0 ≤ x ≤ 1, 0 ≤ y ≤ 2π under the mapping f (z) = e z . (Also, see exercise 3 of chapter 7.) In many cases, however, we can determine the image of the boundary by using the following theorem. Rigidity of Analytic Arcs An arc γ : I → C (I being a real interval) is said to be analytic if γ is the restriction to I of a function γ˜ which is analytic on an open set of C containing I . 14.5 Theorem Let γ :I → C be an analytic arc where I is a compact real interval; let ⊂ C be a circle or a line. If γ [I ] ∩ is infinite, then γ [I ] ⊂ . Note that a very straightforward proof can be given for the case where is a line. By making a simple change of variables, we can assume that is a subset of the real line. In that case, Im γ (t) is a “real” analytic function for t ∈ I. Since Im γ (t) has infinitely many zeroes in I, it follows from the uniqueness theorem for real analytic functions that Im γ (t) is identically zero and γ (t) ∈ R for all t ∈ I. A modified form of the proof can also be given for the case where is a circle. The following argument, however, is applicable to both lines and circles and highlights the fact that the theorem is really about analytic arcs. Proof of Theorem 14.5. Let γ˜ be analytic on some domain containing I with γ˜ | I = γ ; we may arrange that be symmetric about R. Let ( )† denote reflection across and consider ω(z) = (γ˜ (z))† for z ∈ . By the Schwarz reflection principle and by the symmetry of , ω (like γ˜ ) is analytic in . For t ∈ I with γ (t) ∈ , ω(t) = γ (t). But γ (t) ∈ for infinitely many values of t, so that according to the uniqueness theorem (6.10), ω ≡ γ˜ throughout . In particular, for all t ∈ I , γ (t) = γ˜ (t) = ω(t) = (γ (t))† , which implies that γ [I ] ⊂ .


207

14.6 Corollary If f is an analytic mapping of the closed upper half-plane, and if the boundary of its image contains a line segment J , then the boundary of the image is a subset of the line containing J . Proof Note that the boundary of the image is a subset of the analytic curve f (R). If the boundary includes a line segment J , a standard set-theoretic argument shows that, for sufficiently large N, f ([ − N, N]) contains infinitely many points of J . But then, according to Theorem 14.5, the boundary is entirely contained in the line determined by J . It is a well-known consequence of the maximum modulus principle that any nonconstant function g which is C-analytic in D, mapping D onto D and S into S is of the form g(z) = eiθ

n " z − αk for some n ∈ N, αk ∈ D, and θ ∈ R. 1 − αk z

(1)

k=1

(See the solution to Chapter 7, exercise 5.) According to Theorem 14.5, if g is analytic in D, the condition that g maps S into S can be dispensed with. Thus we have 14.7 Corollary An analytic function mapping D onto D is of the form (1). Proof Given such an analytic function f , it suffices to show that f [S] = S. Note that S = ∂( f [D]) ⊂ f [S] by the open mapping theorem. Thus f [S] ∩ S = S. Theorem 14.5 then shows that the analytic arc f [S] is a subset of S. Hence f [S] = S. 14.8 Corollary If an entire function f maps D onto D, then f (z) = cz n for some c ∈ S. Proof

3 Clearly f [D] = D. By Corollary 14.7, f (z) = eiθ nk=1 (z − αk )/(1 − αk z) for some αk ∈ D and θ ∈ R. Since f is entire, all the points αk must be zero, so that f (z) = eiθ z n . It follows that any entire function f mapping some disc onto a disc is of the form f (z) = α(z − z 0 )n + w0 for some α, z 0 , w0 ∈ C and n ∈ N.

208


Analytic Mappings Between Polygons Does there exist any analytic function which maps a closed convex n-gon onto another closed convex n-gon? Of course, if the polygons are similar, there is an elementary linear mapping between them. But what if they are not similar? The somewhat surprising answer is a very general NO, even if all the angles are identical, as would be the case with two dissimilar rectangles. In fact, we begin our inquiry by first considering rectangles. We then answer the question for the smallest n (i.e., n = 3) and finally address the general problem by induction on n. We note at the end of this section that the convexity condition on the polygons is actually not necessary. 14.9 Theorem An analytic function f that maps a closed rectangle R onto another closed rectangle S is a linear polynomial. Theorem 14.5 will play a crucial role here, allowing us to show that f [∂ R] = ∂ S. Proof Assume without loss of generality that R = [0, a] × [0, b] and S = [0, c] × [0, d]. First, note that f will not turn a straight line into an angle unequal in measure to an integral multiple of π, because, where f = 0, f is conformal, and, where f = 0, f magnifies angles by an integral factor. Therefore, f will not map any nonvertex point on ∂ R to a vertex of S. We thus make the following observation: Each vertex of S has precisely one preimage, which is a vertex of R. Hence, the image of each vertex of R must be a vertex of S and f gives a one-to-one correspondence between the vertices of R and those of S. By the open mapping theorem, no interior point of R will be mapped by f to ∂ S, and therefore ∂ S ⊂ f [∂ R]. It is a simple set-theoretic matter that there is a side of ∂ R such that f [] ∩ [0, c] is infinite. Theorem 14.5 then implies that f [] ⊂ [0, c]. Without loss of generality, we may assume that = [0, a], in which case either f (0) = 0 and f (a) = c, or f (0) = c and f (a) = 0. But the latter possibility cannot happen, because f , being analytic, must preserve the orientation of the boundary. It then follows from the intermediate value theorem that f [] = [0, c]. Similarly we conclude that the side {c + i s : s ∈ [0, d]} ⊂ S must be the image of some side of R, and, since f (a) = c, that side of R must be {a +ir : r ∈ [0, b]} ⊂ R. Continuing until we exhaust all four sides of S, we then have established: f maps each side of R onto a side of S. Finally, reflection across the sides of R allows us to extend f (by the Schwarz reflection principle) to rectangles adjacent and congruent to R. Continuing this reflection process, we obtain an entire function, which by construction has at most linear growth in modulus. Hence, by the Extended Liouville Theorem (5.11), f must be a linear polynomial.


209

We note those elements of the preceding argument that are applicable in general. 14.10 Lemma Suppose an analytic function f maps some closed n-gon R onto a closed n-gon S. Then: (a) Each vertex of S has precisely one preimage, which is a vertex of R, and f maps each side of R monotonically onto a side of S. (b) If R and S are both convex, each interior angle of R has the same measure as the corresponding interior angle of S. Proof For part (a), given the argument for Theorem 14.9, we only need to establish monotonicity of f on each side of R. Let be a side of R. If f (z) were to reverse direction as z traverses , there would be some μ ∈ with f (μ) = 0. At the critical point μ, f magnifies angle by an integral factor. Hence the image of any μ-centered semidisc contained in R would lie partly outside of S, contradicting our hypothesis! To show part (b), denote by m(V ) the measure of an interior angle with vertex V . For any vertex A of R, let A = f (A). Then, by analyticity of f , m(A ) = k A m(A) for some k A ∈ N (and, by convexity of S, k A m(A) < π). Since R and S are both convex n-gons,

m(A) = m(A ) = k A m(A). A

A

A

Hence k A = 1 for every vertex A.

We turn now to the general case. The strategy will be to first solve the problem for triangles and then to use induction on the number of sides of the polygons in question to prove the general case. 14.11 Lemma If an analytic function f maps a closed n-gon R onto a closed n-gon S, then f gives ˚ a conformal equivalence between their interiors: R˚ and S. It is interesting that it is the isolated singularities (i.e., vertices) on ∂ R and ∂ S that force f to be conformal. Were the boundaries analytic Jordan curves, no conclusion about the valence of f on R could be drawn (as can be seen by considering z n on the unit disc). Proof By Lemma 14.10(a), f [∂ R] = ∂ S and f |∂ R :∂ R → ∂ S is univalent. Thus the winding number of f |∂ R around any point in S˚ is exactly one. By the argument principle, f ˚ is a conformal equivalence between R˚ and S. We now apply Lemma 14.11 to resolve the triangle case.

210


14.12 Lemma An analytic function f that maps some closed triangle R onto another closed triangle S is a linear polynomial. Proof By Lemma 14.10(b), the two triangles R and S have equal corresponding angles and are therefore similar. We then can easily construct an affine map g mapping R onto S, which is a fortiori conformal. If R is not equilateral, g is unique; if R is equilateral, let g be such that it agrees with f on the vertices of R. By Lemma 14.11, f gives a ˚ Since f and g agree on the three vertices conformal equivalence between R˚ and S. of R, by Proposition 14.4, f = g! Finally, we are able to completely answer the question raised at the beginning of this section. 14.13 Theorem An analytic function mapping some closed convex n-gon R onto another closed convex n-gon S is a linear polynomial. Proof If R is a rectangle, then, according to Lemma 14.10(b), S is also a rectangle. By Theorem 14.9, f is a linear polynomial. Suppose then that R is a nonrectangular quadrilateral. Note that there is at least one interior angle of R that is obtuse. Denote by 1 and 2 the two sides of R that form this angle. Reflect R across 1 and denote by R the reflected image of R. Extend 2 to a full line L 2 . Then L 2 divides R into two regions, one of which is a triangle. Call this triangle T . See the diagram below.

T

R'

l1 l2 R

Observe that, by the Schwarz reflection principle, f can be analytically continued to R and this extended map (which we also call f ) maps R onto the similarlyconstructed quadrilateral S . Since f [2 ] is a side of S, by Theorem 14.5, f [L 2 ∩ R ] is also a line segment. Therefore f [∂ T ] is the boundary of a triangle ; by considering


211

˚ Hence f [T ] = , and, the winding number of f |∂T , we deduce that f [T˚ ] = . according to Lemma 14.12, f must be a linear polynomial. To complete the proof for n > 4, we proceed as in the quadrilateral case and use induction on n. Given a convex n-gon R with n > 4, there must be an interior angle that is obtuse. Otherwise, the formula for the sum of interior angles would be violated. Reflect R across one of the two sides forming this obtuse angle and extend the other side to divide R (the reflected image of R) into two regions, one of which is a convex polygon T with fewer than n sides. As in the quadrilateral case, f extends analytically to R and maps T onto another convex polygon; it is immediate that the polygon f [T ] has the same number of sides as T . The induction hypothesis then guarantees that f is a linear polynomial. In the rectangle case, we first showed that the analytic function in question can be extended to an entire function, and then used its order of growth to prove that it is linear. In the case of other convex polygons, we showed that the analytic function in question can be analytically continued to a mapping between two triangles, which allowed us to conclude that it is a linear polynomial. In both cases, the Schwarz reflection principle and Theorem 14.5 played pivotal roles. Note also that the convexity condition on the polygons in Theorem 14.13 can be dispensed with. This slightly more general result will follow easily. The idea is that, by extending the sides forming an interior angle greater than a straight angle, we can find a convex polygon that is mapped analytically to a convex polygon. We leave the details to the interested reader. Conformal Mappings Between Dissimilar Rectangles Among entire functions, linear polynomials are the only ones that are conformal on every domain. Since a linear polynomial is a composition of rotation, real multiplication, and translation, the two rectangles in Theorem 14.9 and the two n-gons in Theorem 14.13 are actually similar. Let R and S be two closed rectangles that are not similar to each other and let f be a conformal map from R˚ onto S˚ (whose existence is guaranteed by the Riemann mapping theorem). By the Carathéodory-Osgood theorem, f extends to a homeomorphism f˜:R → S. The argument in the proof of Theorem 14.9 shows that • the extension f˜ of f to ∂ R fails to be analytic at some point on ∂ R; • at least one vertex of R fails to be mapped by f˜ to a vertex of S. We can attempt to make f˜ “as analytic as possible” by requiring that three of the four vertices of R be mapped to vertices of S; then by the Schwarz reflection principle, f˜ will be analytic on the two sides of R bounded by the three vertices. Note that, according to Proposition 14.4, the requirement that three chosen vertices of R be mapped to three chosen vertices of S uniquely determines the conformal ˚ One can express such a map explicitly with the aid equivalence f between R˚ and S. of the Schwarz-Christoffel formulae, and this will offer an illustration of the fact that

212


the interiors of two dissimilar rectangles cannot be put into a conformal equivalence that extends to a vertex-preserving homeomorphism. To focus on a simple concrete example, let R = [ − 1, 1] × [0, r ] and S = [ − 1, 1] × [0, s]. Let H denote the upper half-plane and let FR : H → R be the Riemann map with F˜ R (−1) = −1,

F˜ R (0) = 0,

F˜ R (1) = 1.

(1)

Then FR is given by the elliptic integral z 1 FR (z) = Cr dξ 2 (1 − ξ )(1 − kr2 ξ 2 ) 0 1 where kr = 1/ar and Cr = 1/ 0 [(1 − ξ 2 )(1 − kr2 ξ 2 )]−1/2 dξ . It follows from the definition of FR that ar = F˜ R−1 (1 + ir ) increases monotonocally with r and that F˜ R (±ar ) = ±1 + ir. In the same manner, let FS : H → S be the Riemann Mapping, also satisying conditions (1) and with as similarly defined. Suppose f : R˚ → S˚ is a conformal equivalence with f˜(−1) = −1,

f˜(1) = 1,

f˜(1 + ir ) = 1 + i s.

We will see that, if r = s, f˜(−1 + ir ) = −1 + i s and hence f˜−1 (−1 + i s) is not a vertex of R, which implies that f˜ cannot be analytic at f˜−1 (−1 + i s). To this end, we consider g = FS−1 ◦ f ◦ FR : H → H. See the diagram below. f˜

FR H

–1

(–1 + is)

–1 + ir

1 + ir R

–ar

–1

0

1

ar

–1

1

0

g

f FS

–1 + is

1 + is

H f˜ (–1 + ir)

˜ g(–a r) –as

–1

0

1

as

–1

S 0

1

Exercises

213

The map g is an automorphism of the upper half-plane. Hence it is a bilinear mapping with g(−1) ˜ = −1, g(1) ˜ = 1, g(a ˜ r ) = as , with 1 < ar , as .

(2)

If f˜(−1 + ir ) = −1 + i s, we would also have g(−a ˜ r ) = −as . Since the cross-ratio is preserved under bilinear transformations, it would follow that (−ar , −1, 1, ar ) = (−as , −1, 1, as ), or equivalently: (as + 1)2 (ar + 1)2 = 4ar 4as

(3)

It is easy to see, however, that (3) is possible if and only if either ar = as or ar as = 1. The latter equation is impossible, according to (2). Hence ar = as and we have the desired conclusion: f maps vertices of R to vertices of S only if R and S are similar. If r < s, then ar < as . So, if we let g(−a ˜ r ) = −b, by the preservation of crossratios, b < ar . This translates into the sketch of the conformal mapping f in the above diagram. Note that the top side of R is bent by f˜ into an L-shaped path and that f˜ fails to be analytic at precisely one point, i.e., at f˜−1 (−1 + i s). Finally, observe that, despite the fact that f [R] is a rectangle, the image of any subrectangle Q ⊂ R˚ must not be a rectangle; for, otherwise, f would map Q onto a rectangle and thus have to be a linear polynomial, in which case R would necessarily be similar to S.

Exercises 1. Supposeg represents a locally irrotational and source-free flow in a simply-connected domain D and ¯ )dζ . Show that g is orthogonal to the curves given by ReF(z) = constant. F(z) = zz g(ζ 0

2. If F and g are as in (1), show that the curves ImF(z) = constant are the “streamlines” of g; i.e., show that the flow is tangent to those curves. 3. Find the streamlines of the flow functions given by a. g(z) = z¯ b. g(z) = 1/¯z , z = 0. 4. Verify directly that F(z) = z + 1/z is the unique conformal mapping (up to an additive constant) of |z| > 1 onto the exterior of a horizontal interval, with F(z) ∼ z at ∞ [Hint: Begin with the Laurent Expansion A A F(z) = z + A0 + 1 + 22 + · · · z z and use the fact that ImF(eiθ ) = constant (see Markushevich, p. 189).]

214


5. a. Show that w = 2z + 1/z maps the exterior of the unit circle conformally onto the exterior of the ellipse: x2 + y 2 = 1. 9 b. Find a conformal mapping of the exterior of the ellipse x 2 /9 + y 2 = 1 onto the exterior of a real line segment. 6. Given a conformal mapping f of R onto U (the unit disc) and z 0 ∈ R, find a conformal mapping g of R onto U with g(z 0 ) = 0 and g (z 0 ) > 0. 7.* Let R be a simply connected region = C, which is symmetric with respect to the real axis; and suppose that f is the Riemann mapping of R onto U , with f (z 0 ) = 0, f (z 0 ) > 0, for some real-valued z 0 ∈ R. Prove that f (z) = f (z) for all z ∈ R. 8.* Find the unique conformal mapping of the upper half-plane onto the unit disc with a. f (−1), f (0), f (1) equal to 1, i, and −1, respectively. b. f (i) = 0 and f (1) = 1. 9. Let R be simply-connected and assume z 1 , z 2 ∈ R. Show there exists a conformal mapping ofR onto itself, taking z 1 into z 2 . (Consider two cases: R = C and R = C.) 10. Suppose R is any simply connected domain = C. Show that there exists no conformal mapping of C onto R. 11. Let R be a simply connected region and z 0 ∈ R. Suppose G is defined as the set of all analytic functions g : R → U such that g (z 0 ) > 0 (g need not be 1-1). a. Show that sup g (z 0 ) = M ∗ < ∞.

g∈G

b. Assuming that (z 0 ) = M ∗ , show that is 1-1 in R. [Hint: Show that is the Riemann mapping function.] 12.* a. Find a conformal mapping f from the semi-disc S = {z : |z| < 1, Imz > 0} onto the unit disc U and show that it extends to a homeomorphism between S and U . b. Show that f is analytic on S but f −1 is not analytic on U . 13.* Prove that there is no analytic mapping from U onto a “Norman window”, which is a closed region whose boundary is a rectangle surmounted by a semicircle.

Chapter 15

Maximum-Modulus Theorems for Unbounded Domains

15.1 A General Maximum-Modulus Theorem The Maximum-Modulus Theorem (6.13) shows that a function which is C-analytic in a compact domain D assumes its maximum modulus on the boundary. In general, if we consider unbounded domains, the theorem no longer holds. For example, f (z) = e z is analytic and unbounded in the right half-plane despite the fact that on the boundary |e z | = |eiy | = 1. Nevertheless, given certain restrictions on the growth of the function, we can conclude that it attains its maximum modulus on the boundary. The most natural such condition is that the function remain bounded throughout D. 15.1 Theorem Suppose f is C-analytic in a region D. If there are two constants M1 and M2 such that | f (z)| ≤ M1

for z ∈ ∂ D

| f (z)| ≤ M2

for all z ∈ D

then, in fact, | f (z)| ≤ M1 for all z ∈ D. Proof Without loss of generality, we suppose | f (z)| ≤ 1 on ∂ D. Assuming, then, that | f (z)| ≤ M in D, we wish to prove | f (z 0 )| ≤ 1 for every z 0 ∈ D. We will first prove the theorem in the special case where D is the right half-plane and then extend the proof to a general region. In the case of the right half-plane, fix z 0 ∈ D and consider the auxiliary function h(z) =

f N (z) z+1

where N is a positive integer. By the hypothesis on f, |h(z)| ≤ 1 on the imaginary axis and |h(z)| ≤ M N /R for all z ∈ D such that |z| = R. Thus we have 215

216

15 Maximum-Modulus Theorems for Unbounded Domains

|h(z)| ≤ Max(1, M N /R) on the boundary of the right semi-circle D R = {z ∈ D : |z| ≤ R}. Choosing R > M N and large enough so that z 0 ∈ D R , we conclude |h(z)| ≤ 1 along the boundary of the compact domain D R and hence by the Maximum Modulus Theorem |h(z 0 )| ≤ 1. Thus for each z 0 ∈ D N f (z 0 ) z +1 ≤1 0 or | f (z 0 )| ≤ |z 0 + 1|1/N . If we now let N → ∞, we see | f (z 0 )| ≤ 1 as desired. In the more general case, where D is an arbitrary region, we must replace 1/(z +1) by a function g, analytic in D and such that g(z) → 0 as z → ∞. Such a function is given by f (z) − f (a) g(z) = z−a where a is any fixed point in D. Clearly g, like f , is C-analytic in D (Proposition 6.7). The boundedness of f assures g(z) → 0 as z → ∞ and this, in turn, implies that ¯ |g(z)| ≤ K , some constant, throughout D. Again, we set D R = {z ∈ D : |z| ≤ R}. Setting h(z) = f N (z)g(z), because g → 0 as z → ∞ we may take R large enough so that |h(z)| ≤ K along the boundary of D R . Hence, by the Maximum Modulus Theorem, |h(z 0 )| ≤ K for every z 0 ∈ D. Assuming, then, that g(z 0 ) = 0, we can write K 1/N | f (z 0 )| ≤ , g(z 0 ) and letting N → ∞ yields | f (z 0 )| ≤ 1. Note, finally, that unless f is constant, the zeroes of g form a discrete set (Theorem 6.9); hence, by continuity, | f (z 0 )| ≤ 1 for every z 0 ∈ D.

The above theorem may be used to derive the following stronger form of Liouville’s Theorem. 15.2 Definition Let γ be a path parameterized by γ = γ (t), 0 ≤ t < ∞. We will say that f approaches infinity along γ if, for any positive integer N, there exists a point t0 such that | f (γ (t))| ≥ N for all t ≥ t0 . 15.3 Theorem If f is a nonconstant entire function, there exists a curve along which f approaches infinity.

15.1 A General Maximum-Modulus Theorem

217

Note: An equivalent formulation of Liouville’s Theorem (5.10) is that, for any nonconstant entire function f , there exists a sequence of points z 1 , z 2 . . . such that f (z n ) → ∞ as n → ∞. However, the existence of a curve along which f → ∞ does not immediately follow. If we simply connect the points z 1 , z 2 , . . . successively, we have no control over the behavior of f at the intermediate points. The proof of Theorem 15.3 will depend on judiciously choosing the points z k and the connecting lines so that we can guarantee that f → ∞ along the path thus formed. Proof of Theorem 15.3 Let T1 = {z : | f (z)| > 1} and fix S1 , a connected component of T1 . We will need the following facts about S1 : a. S1 is an open set b. | f (z)| = 1 for z ∈ ∂ S1 c. f is unbounded on S1 . (a) is immediate. To prove (b), we first note that | f (z)| ≥ 1 on the boundary of S1 by continuity. If | f (z)| > 1 for some z ∈ ∂ S1 , then | f (w)| > 1 for all w in a neighborhood of z and thus z would be an interior, rather than a boundary point of S1 . Finally, if f were bounded throughout S1 , we could apply (b) and Theorem 15.1 to show | f (z)| ≤ 1 throughout S1 , contradicting its definition. Now set T2 = {z ∈ S1 : | f (z)| > 2} and choose a connected component S2 . (Note that, by (c), T2 is non-empty.) As above, we can prove that f is unbounded on S2 . Proceeding inductively, we obtain a sequence of regions S1 ⊃ S2 ⊃ S3 ⊃ · · · such that | f (z)| > k for all z ∈ Sk .

z1 S1

γ

z2

S2

z3 S3

Finally, we choose a point z k ∈ Sk for k = 1, 2, . . .. Since each set Sk is a region which contains all points z n , n ≥ k, we can connect z k to z k+1 by a polygonal path 2 γk contained in Sk . Thus | f (z)| > k for all z ∈ γk . If we then form the path γ = ∞ k=1 γk , it follows that f approaches ∞ along γ , proving the theorem.

218

15 Maximum-Modulus Theorems for Unbounded Domains

15.2 The Phragmén-Lindelöf Theorem We now return to theorems of maximum-modulus type. Theorem 15.1 is rather general in that it applies to any region. On the other hand, if we restrict ourselves to various specific regions D, we will be able to derive the same type of conclusion under a much weaker hypothesis on f . We begin, as before, by considering the right half-plane. As previously noted, the function e z is unbounded in this domain despite the fact that it is bounded by 1 on the imaginary axis. The same, of course, is true of the function eδz for any δ > 0. However, if f (z) has slower growth than eδz , we have the following extension of Theorem 15.1. 15.4 Phragmén-Lindelöf Theorem Let D denote the right half-plane and suppose f is C-analytic in D. If | f (z)| ≤ 1

(1)

on the imaginary axis and if, for each > 0, there exists a constant A such that | f (z)| ≤ A e|z|

(2)

throughout D, then (1) holds for all z ∈ D. Before proceeding with the proof, we will need the following lemma, which is a slightly weaker form of the theorem. Lemma 1 Suppose f is C-analytic in the right half-plane D. If | f (z)| ≤ 1

(3)

on the imaginary axis, and if for some δ > 0, there exist constants A and B such that | f (z)| ≤ A exp(B|z|1−δ )

(4)

for all z ∈ D, then (3) holds throughout D. Proof of Lemma 1 Here we use the auxiliary function h(z) =

f N (z) exp(z 1−δ/2 )

and wish to show |h(z 0 )| ≤ 1 for each z 0 ∈ D. Let us first analyze the denominator g(z) = exp(z 1−δ/2 ). In the open right half-plane z 1−δ/2 may be defined as an analytic function (see the comments following Theorem 8.8). To fix its value, we take it to

15.2 The Phragmén-Lindelöf Theorem

219

be positive on the positive real axis. Then, for z = r eiθ , − π/2 < θ < π/2, z 1−δ/2 = r 1−δ/2 eiθ(1−δ/2) , −

π π 0, there exist constants A and B such that | f (z)| ≤ A exp(B|z|2−δ ) for every z in the quadrant, then | f (z)| ≤ 1 throughout the quadrant. Proof of Lemma 2

√ Without loss of generality, we consider the first quadrant. Set g(z) = f ( z). Then g is C-analytic in the upper half-plane. Furthermore, by the hypothesis on f, |g(z)| ≤ 1 on the boundary and |g(z)| ≤ A exp(B|z|1−δ/2) throughout the half-plane. By Lemma 1, |g(z)| ≤ 1 throughout the half-plane and thus | f (z)| ≤ 1 for all points z in the quadrant. Proof of Theorem 15.4 We consider

f N (z) ez and, as before, the proof will follow if we can show |h(z)| ≤ 1 throughout the right half-plane. To do this, we consider the first and fourth quadrants separately. To estimate h(z) on the boundary of the first quadrant, note that |eiy | = 1 and hence, by (1) |h(z)| ≤ 1 on the positive imaginary axis. h(z) =

Also, by (2), | f (z)| ≤ A1/N e(1/N)|z| so that setting B N = (A1/N ) N , |f ≤ B N e|z| throughout the half-plane. On the positive x-axis, though, z |z| |e | = e and hence |h(z)| ≤ B N for z > 0. Thus |h(z)| ≤ Max(1, B N ) along the boundary of the first quadrant. Furthermore, throughout the first quadrant N (z)|

|h(z)| ≤ | f N (z)| ≤ B N e|z| so that we can apply Lemma 2 to conclude |h(z)| ≤ Max(1, B N ) in the first quadrant. By the exact same reasoning, |h(z)| ≤ Max(1, B N ) in the fourth quadrant. Hence h(z) is a bounded C-analytic function in the right half-plane and is bounded by 1 on the imaginary axis. By Theorem 15.1, |h(z)| ≤ 1 throughout the right half-plane, and the proof is complete.

15.2 The Phragmén-Lindelöf Theorem

221

By mapping a wedge of angle α onto the right half-plane, we derive the following corollary. 15.5 Corollary α α D = z : − < Arg z < , where 0 < α ≤ 2π, 2 2 and suppose f is C-analytic in D. If Let

| f (z)| ≤ 1

(1)

on ∂ D and if, for each > 0, there exists a constant A such that | f (z)| ≤ A exp(|z|π/α ),

(2)

then (1) holds throughout D. Proof Given f as above, consider g(z) = f (z α/π ) in the right half-plane and apply Theorem 15.4. An interesting special case of the corollary arises if we take a wedge of angle 2π (the whole plane slit along one ray). In that case, the boundary is a single ray and, √ by the above corollary, if f is bounded on that ray and has slower growth than e |z| for each > 0, it is in fact bounded throughout the wedge. Now, we may view an entire function as a C-analytic function in every wedge of angle 2π. This leads to the following theorem. 15.6 Theorem If f is a non-constant entire function and for each > 0 there exists a constant A such that √ | f (z)| ≤ A e |z| then f (z) is unbounded on every ray! Proof If f were bounded on some ray R, by Corollary 15.5 it would also be bounded on the wedge C\R; that is, f would be bounded in the entire plane. But, then, by Liouville’s Theorem f would reduce to a constant, contradicting the hypothesis of the theorem. E XAMPLE √ cos z has a power series involving only even terms, hence cos z is an entire function that is bounded on the positive x-axis. Hence, by the above theorem, it must grow

222

as fast as

15 Maximum-Modulus Theorems for Unbounded Domains √ e |z|

for some > 0. Setting eiz + e−iz 2

cos z =

♦

shows that this is in fact the case. (Consider points z along the imaginary axis.)

An Application of Theorem 15.6: The differential equation f (z) = − f (z) has the explicit solution f (z) = Ae−z . However, if we consider the very similar equation z (1) f (z) = − f 2 no such explicit solution can be found. Nevertheless, one may seek to study the behavior of a solution f (z) as z → ∞ along the positive x-axis. To accomplish this, we will find the solution in the form of a power series which is, in fact, an entire function. Furthermore, we will show that the solution is of “small” growth, so that Theorem 15.6 is applicable, and f is unbounded on every ray. Thus, unlike Ae−z , the solution to (1) has no limit as z → +∞. The details are as follows: 15.7 Proposition Let f be a solution of the differential equation f (z) = − f (z/2), analytic at z = 0. Then f is entire and is unbounded on every ray. Proof Let f have the power series representation f (z) =

∞

ak z k .

k=0

Because of (1), we must have ∞

kak z k−1 = −

k=1

∞

ak

z k

k=1

or ak = − By induction, then, an =

2

ak−1 . 2k−1 k

(−1)n a0 . n!2n(n−1)/2

Hence f (z) is given by f (z) = A

k

bk z k

where bk =

(−1)k , k!2k(k−1)/2

(2)

and a simple check shows that (2) does in fact represent an entire solution of (1).

Exercises

223

We now show that f satisfies the hypothesis of Theorem 15.6. For this we fix > 0 and show that, for z sufficiently large, | f (z)| < exp(|z| ). Assume then that |z| = R = 2 N , N > 2, and let M(R) = Max | f (z)|, |z|=R

M = M(1).

According to (1), f (z) − f (0) =

z

f (ρ)dρ = 0 R RM 2

so that | f (z)| ≤ 2RM

z

−f

0

ρ 2

dρ

R . 2

Setting M(R/2) = | f (z 1 )| for some z 1 ∈ D(0; R/2) and proceeding inductively, we obtain | f (z)| ≤ M R N = M|z|| log z|/ log 2 . (3) The right-hand side of (3) is bounded above by exp(|z| ) for all z sufficiently large; therefore, we get R0 = R0 () such that | f (z)| < exp(|z| ) for all z with |z| ≥ R0 , as desired.

Exercises 1.* Show that the conclusion of Theorem 15.1 would hold if we insisted only that f 1 along the boundary and f (z) log z throughout the domain. How could the hypothesis be further relaxed? 2. What is the “smallest” non-constant analytic function in the quadrant D = {x + iy : x, y < 0}, which is bounded along the boundary? z

3. Show that ee 1 throughout the boundary of the region π π . D = x + iy : − < y < 2 2 Show that it is the “smallest” such analytic function. 4.* Suppose g is a non-constant entire function which is bounded on every ray. (See 12.2). Show that for any A and B, there must exist some point z with |g(z)| > A exp(|z| B ). [Hint: If not, divide the plane into a finite number of very small wedges and apply 15.5 and Liouville’s Theorem to conclude that g is constant.]

Chapter 16

Harmonic Functions

16.1 Poisson Formulae and the Dirichlet Problem In this chapter, we focus on the real parts of analytic functions and their connection with real harmonic functions. 16.1 Definition A real-valued function u(x, y) which is twice continuously differentiable and satisfies Laplace’s equation u x x + u yy = 0 throughout a domain D is said to be harmonic in D. Although one may talk of complex-valued harmonic functions, the term “harmonic” throughout this chapter will always refer to a real-valued function. 16.2 Theorem If f = u + i v is analytic in D, u and v are harmonic there. Proof u and v both have continuous partial derivatives of all orders since f is analytic. By the Cauchy-Riemann equations ux = vy;

u y = −νx

so that u x x = v yx = v x y = −u yy , hence u is harmonic. By the same argument, v is harmonic since it is the real part of the analytic function −if. The converse of the above is not true. For example, u(x, y) = log(x 2 + y 2 )

225

226

16 Harmonic Functions

is harmonic in the punctured plane but is not the real part of an analytic function there. (See Exercise 4.) We do have the following partial converse: 16.3 Theorem If u is harmonic in D, then a. u x is the real part of an analytic function in D; b. if D is simply connected, u is the real part of an analytic function in D. Proof a. Let f = u x − i u y . Since u ∈ C 2 , f has continuous first-order partial derivatives. Moreover, by the harmonicity of u f y = u x y − iu yy = u yx + iux x = ifx so that f satisfies the Cauchy-Riemann equations. Hence f is analytic in D. b. If D is simply connected, by the Integral Theorem (8.5), f = u x − i u y is the derivative of an analytic function F. But then if F = A + i B F (z) = A x + iBx = A x − iA y = u x − iu y so that A(x, y) = u(x, y) + C. Hence u(x, y) is the real part of the analytic function F(z) − C.

E XAMPLE u(x, y) = x − e x sin y is harmonic in the whole plane. Hence f (z) = u x (z) − i u y (z) = 1 − e x sin y + iex cos y is entire. In fact, f (z) = 1 + iez and if we set z F(z) = f (ζ )dζ = z + iez − i, 0

then u(z) = ReF(z).

♦

The fact that a harmonic function is, at least locally, the real part of an analytic function allows us to apply some of the theory of analytic functions to harmonic functions. 16.4 Mean-Value Theorem for Harmonic Functions If u is harmonic in D(z 0 ; R), 1 u(z 0 ) = 2π for all positive r < R.

2π 0

u(z 0 + r eiθ )dθ

16.1 Poisson Formulae and the Dirichlet Problem

227

Proof Let u = Re f . By 6.12 1 f (z 0 ) = 2π

2π

f (z 0 + reiθ )dθ,

0

and the result follows by taking the real parts of the above.

16.5 Maximum-Modulus Theorem for Harmonic Functions If u is a nonconstant harmonic function in a region D, u has no maximum or minimum points in D. Proof The theorem may be derived as a corollary of the above Mean-Value Theorem. It follows even more immediately, however, from the Open Mapping Theorem (7.1). For in a disc D(z 0 ; δ) ⊂ D about any point z 0 ∈ D, u is the real part of an analytic function f . Since f maps D(z 0 ; δ) onto an open set, u takes both larger and smaller values than u(z 0 ) in the open disc. Note that the Maximum-Modulus Theorem for analytic functions (6.13) asserts only that | f | has no interior maximum point; | f | can have a local minimum if it is equal to zero. By contrast, Theorem 16.5 shows that a non-constant harmonic function has neither a maximum nor a minimum point in the interior of a domain. We let the term C-harmonic refer to a function which is harmonic in the interior of a domain and continuous on the closure. The previous theorem implies then that a C-harmonic function in a compact domain must assume its maximum and minimum values on the boundary of that domain. 16.6 Corollary If two C-harmonic functions u 1 and u 2 agree on the boundary of a compact domain D, then u 1 = u 2 throughout D. Proof u = u 1 − u 2 is C-harmonic in D; hence it takes its maximum and minimum on the boundary. Since u ≡ 0 on the boundary, it follows that u ≡ 0 throughout D and that u 1 ≡ u2. Corollary 16.6 shows that a C-harmonic function is determined by its values on the boundary of a compact domain. But this result is of a purely theoretical nature. How to determine the value at an interior point from a knowledge of u on the boundary is the subject of the next theorem. We begin by considering C-harmonic functions in the unit disc.

228


16.7 Theorem Suppose u is C-harmonic in D(0;1). Then 1 u(z) = 2π

2π

u(eiθ )K(θ, z)dθ

0

where K(θ, z) is the “Poisson Kernel,” eiθ + z . K(θ, z) = Re iθ e −z

In polar form, 1 u(re ) = 2π

2π

iϕ

0

u(eiθ )(1 − r 2 ) dθ. 1 − 2r cos(θ − ϕ) + r 2

Proof [To simplify the notation, we will assume u = Re f where f is analytic on the closed unit disc. To justify the assumption, we could first prove the theorem for u ∗ (z) = u(r z) where r < 1 and then take the limit as r → 1 since u is uniformly continuous on D(0; 1).] By the Cauchy Integral Formula (6.4) f (ζ ) 1 f (z) = dζ 2πi |ζ |=1 ζ − z or iθ 2π 1 e iθ f (z) = dθ (1) f (e ) iθ 2π 0 e −z If we replace z by the symmetric point 1/¯z which lies outside the unit disc, then by the Closed Curve Theorem (8.6) 1 f (ζ ) 0= dζ 2πi |ζ |=1 ζ − 1z¯ or 1 0= 2π

0

2π

eiθ dθ. f (e ) iθ e − 1/¯z

iθ

Note that z¯ eiθ −¯z eiθ = = −iθ iθ iθ e − 1/¯z z¯ e − 1 e − z¯ −iθ eiθ e = 1 − iθ , = 1 − −iθ e − z¯ e −z

(2)


229

so that subtracting (2) from (1) yields iθ 2π 1 e eiθ iθ + iθ − 1 dθ f (e ) iθ f (z) = 2π 0 e −z e −z iθ 2π 1 e = − 1 dθ, f (eiθ ) 2 Re iθ 2π 0 e −z or

1 f (z) = 2π

2π 0

eiθ + z dθ. f (e )Re iθ e −z

iθ

Finally, taking the real parts of the above, we obtain iθ 2π e +z 1 iθ dθ. u(e )Re iθ u(z) = 2π 0 e −z

(3)

(4)

By mapping the unit disc onto other domains, we can obtain similar results for any simply connected domain. For example, if u is harmonic in D(0; R), u = Re f , we can apply the above results to g(ζ ) = f (Rζ ). Thus iθ 2π 1 e +ζ f (Rζ ) = dθ, f (Reiθ )Re iθ 2π 0 e −ζ and if we let Rζ = z f (z) =

1 2π

and u(z) =

1 2π

2π

f (Reiθ )Re

0

0

2π

u(Reiθ )Re

Reiθ + z dθ, Reiθ − z

(5)

Reiθ + z dθ. Reiθ − z

(6)

The above is known as the Poisson Integral Formula for a disc. The Poisson Formula for a bounded harmonic function in a half-plane 1 ∞ yu(t) u(x + i y) = dt (7) π −∞ (t − x)2 + y 2 is derived in Exercise 6. The Dirichlet Problem The Dirichlet Problem is the problem of proving the existence of a function u which is C-harmonic in a domain and assumes prescribed boundary values. This differs from the attitude in the last section where a function u was assumed to be C-harmonic in a domain and we sought a formula for u in terms of its boundary values. Nevertheless, the previous theorems offer a starting point. Suppose, for example, that D is the unit disc. Then if there is a harmonic function u

230


in D with limit values u(eiθ ) on the boundary, u must be of the form 1 u(z) = 2π or u(reiϕ ) =

1 2π

2π 0

2π 0

eiθ + z dθ, u(e )Re iθ e −z

iθ

u(eiθ )(1 − r 2 ) dθ. 1 − 2r cos(θ − ϕ) + r 2

The fact that this Poisson Integral does indeed provide the solution to the Dirichlet Problem is proven below. 16.8 Theorem Suppose u(eiθ ) is continuous on C(0; 1). Then 1 u(z) = 2π

2π

u(eiθ )K(θ, z)dθ

0

is the restriction to D(0; 1) of a C-harmonic function in the closed unit disc with boundary values u(eiθ ). Proof Let g(z) = (eiθ

1 2π

2π

u(eiθ )

0

eiθ + z dθ, |z| < 1. eiθ − z

Since + − z) is an analytic function of z for each θ and since g is continuous, it follows by Morera’s Theorem that g is analytic in D(0; 1). Moreover, u(z) = Re g(z) so that u is harmonic. To show that u has the limit u(eiθ ) as z → eiθ , we note the following properties of the Poisson Kernel z)/(eiθ

K(θ, z) =

1 − r2 , z = r eiϕ . 1 − 2r cos(θ − ϕ) + r 2

(8)

i. K(θ, z) > 0. The numerator is obviously positive and the denominator is bigger than (1 − r )2 . 2π ii. (1/2π) 0 K(θ, z)dθ = 1. This follows on applying the Poisson Formula (16.7) with u ≡ 1. iii. For every δ > 0 ϕ−δ

0

K(θ, z)dθ +

2π

ϕ+δ

K(θ, z)dθ → 0

as z → eiϕ .

Note that the denominator in (8) is bounded away from zero for |θ − ϕ| > δ while the numerator approaches 0 as z approaches the boundary.


231

According to (ii) we can write u(reiϕ ) − u(eiϕ ) =

1 2π

2π

[u(eiθ ) − u(eiϕ )]K(θ, z)dθ.

0

Let M = Maxθ |u(eiθ )|. By the continuity of u, given > 0 we can find δ > 0 so that |u(eiθ ) − u(eiϕ )| < when |θ − ϕ| < δ. Then by (ii) and (iii) 2π ϕ−δ M K(θ, z)dθ + K(θ, z)dθ |u(reiϕ ) − u(eiϕ )| ≤ π 0 ϕ+δ ϕ+δ 1 + K(θ, z)dθ 2π ϕ−δ 2π ϕ−δ M K(θ, z)dθ + K(θ, z)dθ + , ≤ π 0 ϕ+δ and as r → 1

limr→1 |u(reiϕ ) − u(eiϕ )| ≤ .

Hence lim u(reiϕ ) = u(eiϕ ).

r→1

(9)

Since u was assumed to be continuous on the unit circle and u is harmonic (and hence continuous) in the disc, it follows from (9) that u is continuous in D¯ with the prescribed values on the boundary.

Remarks 1. According to Corollary 16.6, the above solution to the Dirichlet Problem is unique. 2. The arguments above show that for any integrable function u on the unit circle, there is a harmonic function in D(0; 1) with limit u(e iϕ ) at any point of continuity of u along the boundary. 3. By considering the appropriate conformal mapping f of D onto U , we can solve the Dirichlet Problem for any bounded simply connected domain. To find a harmonic function u 1 in D with given boundary values, we first determine a harmonic function u 2 in U with the values u 1 ( f −1 (z)) along the boundary. Since u 2 is the real part of an analytic function g, u 1 (z) = u 2 ( f (z)) = Re g( f (z)) is the desired harmonic function in D. 4. In many simple cases, an explicit solution to the Dirichlet Problem can be obtained (without recourse to the Poisson Integral) by determining an analytic function with the appropriate real part.

232


E XAMPLES i. To determine the C-harmonic function u in D(0; 1) with boundary values u(x, y) = x 2 , note that Re z 2 = x 2 − y 2 is everywhere harmonic and equals 2x 2 − 1 on the boundary. Hence u(x, y) =

1 2 1 (x − y 2 ) + . 2 2

By taking linear combinations of the above with the harmonic polynomials 1, x, y and x y, we can find a C-harmonic function in D(0; 1) with boundary values equal to any given quadratic polynomial on C(0; 1). ii. log r = Re log z is a harmonic function in the punctured plane z = 0 which depends only on the modulus. [Although log z is only analytic in a slit plane, Re log z = log |z| is continuous and hence harmonic in the entire punctured plane.] Thus if A is an annulus: r1 ≤ |z| ≤ r2 , we can find a harmonic function in A with arbitrary constant values on the inner and outer circles by setting u(r eiϕ ) = a log r + b for appropriate a and b. ♦ An Application to Heat Problems. Suppose we consider a solid whose temperature u is constant in one direction. (This is a reasonable model for a cylindrical solid with insulated faces or for a “very long” cylindrical solid.) If we assume that the temperature is independent of time, then, thinking of the solid as resting in a region of the z plane, the temperature depends only on the x, y position and can be shown to be a harmonic function. [See Appendix III.] For that reason, Laplace’s equation: u x x + u yy = 0, is sometimes called the heat equation and Dirichlet problems can be thought of as boundary-value heat problems. Such problems can thus be solved by the methods discussed. It is often helpful to first map the given region onto a simpler one where a solution to the corresponding problem is known. E XAMPLES i. Suppose the annulus 1 ≤ |z| ≤ 2 represents the cross-section of an “infinite” cylindrical solid with temperature 100◦ maintained on the outer rim and temperature 0◦ on the inner rim. Then, as in the previous example, the temperature is given by log r iφ 100◦ . u(r e ) = log 2 √ In particular, the isothermal line with temperature 50◦ is the circle of radius 2. ii. Next we find the “steady-state” temperature function in the unit disc with boundary values 1 on the upper semi-circle and 0 on the lower semi-circle. Note that w = (z − 1)/(z + 1) maps the disc onto the left half-plane with the upper and lower semi-circles mapping onto the positive and negative imaginary axes, respectively. In the left half-plane, Arg z = Im log z is harmonic with boundary values π/2

16.2 Liouville Theorems for Re f ; Zeroes of Entire Functions of Finite Order

233

and 3π/2, so that 3 Arg z − 2 π has the desired boundary values. The solution to the given problem, then, is 3 1 z−1 u(z) = − Arg . 2 π z+1

♦

16.2 Liouville Theorems for Re f ; Zeroes of Entire Functions of Finite Order The following theorem offers a formula, much like the Poisson Integral Formula, for the value of a C-analytic function in D(0; R) in terms of its real part. This, in turn, will allow us to obtain estimates on the magnitude of an entire function from given bounds on its real part alone. 16.9 Theorem If f = u + i v is C-analytic in D(0; R), then iθ 2π 1 Re + z iθ f (z) = dθ + i v(0). u(Re ) 2π 0 Reiθ − z Proof We have already proven (following Theorem 16.7) that if f = u + i v is C-analytic in D(0; R) then iθ 2π 1 Re + z iθ dθ. (1) f (z) = f (Re ) Re 2π 0 Reiθ − z Moreover, as we noted in the proof of 16.8 (with R = 1) iθ 2π 1 Re + z g(z) = dθ u(Reiθ ) 2π 0 Reiθ − z

(2)

is also analytic in D(0; R). A comparison of (1) and (2) shows, however, that f and g have the same real parts Re f (z) = Re g(z) =

1 2π

2π

u(Reiθ ) Re

0

Hence f (z) = g(z) + i λ

Reiθ + z dθ. Reiθ − z

234


or 1 f (z) = 2π

2π 0

Reiθ + z dθ + i λ. u(Re ) Reiθ − z

iθ

To determine λ, set z = 0. Then, by the Mean-Value Theorem (16.4), the integral on the right equals u(0), so that f (0) = u(0) + i λ and λ = v(0).

Analogues of Liouville’s Theorems for Re f . The original Liouville Theorem (5.10) states that a bounded entire function is constant. Note that the condition | f | ≤ M implies the four inequalities −M ≤ Re f ≤ M −M ≤ Im f ≤ M. However, according to the Weierstrass Theorem (9.6), any one of the four inequalities would suffice to prove that f is constant. For if any one of the inequalities is satisfied, the set of values assumed by f is not dense in the whole plane and f must be constant. The next theorem shows that the same reduction in hypothesis is possible for the Extended Liouville Theorem (5.11). 16.10 Theorem If f is entire and any one of the four inequalities −A|z|n ≤ Re f (z) ≤ A|z|n −A|z|n ≤ Im f (z) ≤ A|z|n holds for sufficiently large z, then f is a polynomial of degree ≤ n. Proof Without loss of generality, we may assume Re f (z) ≤ A|z|n for large z. (In the other cases, we could consider − f or if.) Then applying 16.9 with R = 2|z| iθ Re + z Reiθ − z ≤ 3 and | f (z)| ≤

3 2π

2π

|u(Reiθ )|dθ + | f (0)|, where u = Re f.

0

To estimate the integral above, we set u(ζ ) if u(ζ ) > 0 + u (ζ ) = 0 if u(ζ ) ≤ 0.


235

Then according to the hypothesis, if |z| is large enough, 1 2π

2π

u + (Reiθ ) dθ ≤ A R n = A2n |z|n

0

and by the Mean-Value Theorem (16.4)

1 2π

2π

u(Reiθ ) dθ = u(0).

0

By the lemma below 1 2π

2π

|u(Reiθ )| dθ ≤ A2n+1 |z|n + |u(0)|

0

so that | f (z)| ≤ A1 |z|n + A2 and by the Extended Liouville Theorem, f is a polynomial of degree at most n. 16.11 Lemma Let g be real-valued and continuous on [a, b]. If

b

b a

g(x)d x = α and if

g + (x)d x ≤ β,

a

then

b

|g(x)| d x ≤ 2β + |α|.

a

Proof

Recall that +

g (x) = If we set

g(x) if g(x) > 0 0 if g(x) ≤ 0.

−g(x) if g(x) < 0 g (x) = 0 if g(x) ≥ 0, −

then and By hypothesis

g = g+ − g− |g| = g + + g − .

b a

g + (x) ≤ β

236

and


b

−

a

so that

b

g (x) d x =

g + (x) d x − α ≤ β − α,

a

b

|g(x)| d x ≤ 2β − α ≤ 2β + |α|.

a

16.12 Definition An entire function f is said to be of finite order if for some k and some R > 0, | f (z)| ≤ exp(|z|k ) for all z with |z| ≥ R. Theorem 16.10 can be used to prove the existence of zeroes for many entire functions of finite order. To show, for example, that e z − z must have a zero, we first assume that e z − z = 0. Then g(z) = log(e z − z) would be entire with Re(g(z)) = log |e z − z| ≤ |z| + 1 for |z| ≥ e. But then, according to Theorem 16.10, g would be a linear polynomial; that is, log(e z − z) = az + b or e z − z = eaz+b . Expanding both sides in power series would lead us to conclude

2 z3 z2 b 2z + + · · · = e 1 + az + a + ··· , 1+ 2! 3! 2! which is impossible. Similarly, we can show that e z − z must have infinitely many zeroes. For if e z − z had only finitely many zeroes α1 , α2 , . . . , α N , we could apply the above argument to (e z − z)/(z − α1 )(z − α2 ) . . . (z − α N ) to conclude that e z − z = (z − α1 )(z − α2 ) . . . (z − α N )eaz+b .

(3)

By considering the growth of both sides as z → ∞, however, it is easily seen that (3) cannot hold. 16.13 Theorem Suppose f is an entire function of finite order. Then either f has infinitely many zeroes or f (z) = Q(z)e P(z) where Q and P are polynomials.


237

Proof Suppose f has a finite number of zeroes, α1 , . . . , αk . Then we may write f (z) = Q(z)g(z) where Q(z) = (z − α1 ) · · · · ·(z − αk ) and g is an entire function that is never zero. Thus we can define an entire function P(z) = log g(z), which by our hypothesis must satisfy |Re P(z)| ≤ |z|k for |z| ≥ R for some k and R. Hence P is a polynomial and f (z) = Q(z)e P(z) , as desired. An entire function does not have to assume every value in the complex plane. However, according to Theorem 16.13, if f is of finite order, and if f (z) = a for all z, then f (z) − a = e P(z) It follows that f assumes every other complex value b infinitely often since P assumes each of the infinitely many values of log(b − a). The Little Picard Theorem asserts that the above is true for all entire functions. While a proof of this theorem would take us too far afield, we can prove that it is true for a very broad class of functions. Let E 1 (z) = exp(z k ) for any fixed positive integer k, and let E n+1 (z) = exp(E n (z)), so that E j is the j -fold exponential of z k . We will show that Picard’s Theorem is applicable to any entire function which grows no faster than E j for some j. To be precise, we will say that f is of j −fold exponential order if, for some R > 0, the j -fold logarithm: log(log(log · · ·(| f (z)|))) < |z|k , for some fixed k and |z| > R. Note that if f is of j −fold exponential order, then log f is of ( j − 1)−fold exponential order. 16.14 Theorem Suppose f is an entire function of j −fold exponential order, for some j. Then, if f (z) = a for all z, f assumes every other complex value b infinitely often. Proof If j = 1, f is of finite order and the result follows, as indicated above. To complete the proof, assume that f is of ( j + 1)-fold exponential order. Then g(z) = log( f (z) − a) would be of j -fold exponential order, and by induction we can assume that g assumes every value in the complex plane with at most one exception. In particular, g assumes

238


all of the infinitely many values of log(b − a) with at most one exception. But then, since f (z) = a + e g(z), f assumes every complex value b = a infinitely often. We leave it as an exercise to show that Theorem 16.14 is not equivalent to Picard’s Little Theorem. That is, there are entire functions which are not of j−fold exponential order, for any j.

Exercises 1. Suppose f = u + iv is analytic. Show then that u + v and uv are harmonic. 2. Show that every partial derivative of a harmonic function is itself harmonic. 3. Show that u 2 cannot be harmonic for any nonconstant harmonic function u. 4. Show that log(x 2 + y 2 ) is harmonic in z = 0 but is not equal to the real part of a function that is analytic in z = 0. 5. a. Show that if u(r, θ ) is dependent on r alone, Laplace’s equation becomes u rr +

1 u r = 0. r

b. Use the above to show that a harmonic function which depends on r alone must have the form u(r, θ ) = a log r + b. 6. Derive the Poisson Formula u(x + iy) =

1 ∞ y · u(t)dt π −∞ (t − x)2 + y 2

(I)

for a bounded C-harmonic function in the upper half-plane. [Hint: Let C R denote the indicated contour and set f (ζ ) f (ζ ) dζ − dζ, 2π if(z) = ζ − z ζ CR CR − z where Re f = u. Then simplify and obtain (I) for f (x + iy) by letting R → ∞.]

CR z

–R

R –z

7. Find a harmonic function in D(0; 1) with boundary values u(x, y) = x 3 . 8. Let u be harmonic in D(0; 1) with boundary values: 1 on the upper semi-circle and 0 on the lower semi-circle. Show that the level curves u(x, y) = k, 0 ≤ k ≤ 1, are all circular segments.

Exercises

239

9. Find a harmonic function u in the upper half-plane with lim u(x, y) =

y→0

0, x < 0 1, x > 0

10. Find the temperature function u(x, y) for a solid represented by the semi-infinite strip −

π π ≤x ≤ , y≥0 2 2

given that u(x, 0) = 1, − π2 < x < π2 , π π , y = 0, for y > 0. u − , y = 2, and u 2 2 11. Prove ez − P(z) and sin z − P(z) have infinitely many zeroes for every non-zero polynomial P. 12. We say an entire function of finite order has order j if j = inf k : lim

f (z) = 0 . z→∞ exp(|z|k )

Prove that the only non-vanishing entire functions of order j are of the form f (z) = e P j (z) , where P j is a polynomial of degree j. 13.* Show that sin z − z = c has a solution for every complex number c by showing that if sin z − z = c for all z, then sin z − z = c + 2π 14.* Let f 0 (z) = z, f n+1 (z) = e fn (z) , n = 0, 1, 2, ... and let g0 (t) = t, gn+1 (t) = t gn (t) . Define F(z) =

∞

f k (z) gk (k)

k=0

Show that F(z) is an entire function but F(z) is not of j-fold exponential order for any positive integer j.

Chapter 17

Different Forms of Analytic Functions

Introduction The analytic functions we have encountered so far have generally been defined either by power series or as a combination of the elementary polynomial, trigonometric and exponential functions, along with their inverse functions. In this chapter, we consider three different ways of representing analytic functions. We begin with infinite products and then take a closer look at functions defined by definite integrals, a topic touched upon earlier in Chapter 7 and in Chapter 12.2. Finally, we define Dirichlet series, which provide a link between analytic functions and number theory.

17.1 Infinite Products 17.1 Definition a. 3 Let {u k }∞ k=1 be a sequence of nonzero complex numbers. The infinite product ∞ u is said to converge if the sequence of partial products PN = u 1 u 2 . . . u N k k=1 converges to a nonzero limit as N → ∞. If PN → 0, we say the infinite product diverges to 0. b. If finitely many3terms u k are equal to zero, we will say the product converges to zero provided ∞ k=1 u k converges. u k =0

E XAMPLES 3 i. ∞ (1 + 1/k) = 21 · 32 · 43 · . . . diverges (to ∞) since PN = N + 1 → ∞. 3k=1 (1 − 1/k) diverges to zero. ii. ∞ 3 3k=2 ∞ 2 iii. k=2 (1 − 1/k 2 ) = ∞ k=2 (k − 1)(k + 1)/k converges. this by finding an explicit formula for PN . 3 We leave it 2as an exercise to prove 3 ∞ 2 iv. ∞ (1 − 1/k ) converges to 0 since ♦ k=1 k=2 (1 − 1/k ) converges.

241

242

17 Different Forms of Analytic Functions

Notes 1. If PN−1 = 0, uN =

PN . PN−1

3 Hence if ∞ k=1 u k converges, u N → 1 as 3 N → ∞. For this reason, we will usually write infinite products in the form k (1 + z k ) with the understanding that z k → 0 if the product converges. 3 2. If {ak }∞ is a sequence of positive real numbers, ∞ k=1 (1 + ak ) converges if and k=1 ∞ only if k=1 ak converges. This follows from the inequalities a1 + a2 + · · · + a N ≤

N "

(1 + ak ) ≤ ea1 +a2 +···+a N .

k=1

The right-hand inequality is a direct consequence of the fact that31 + x ≤ e x for all real x. It is not true complex numbers z k , however, that ∞ k=1 (1 + z k ) for ∞ converges if any only if k=1 z k converges (see Exercise 5), but we do have the following theorem. 17.2 Proposition

3 ∞ Let z k = −1, k = 1, 2, . . . . ∞ k=1 (1 + z k ) converges if and only if k=1 log(1 + z k ) converges. (log z here denotes the principal branch of the logarithm; i.e., −π < Im log z = Arg z ≤ π.) Proof

N Let SN = k=1 log(1 + z k ). Then PN = e S N and if SN → S, PN → P = e S . Suppose, on the other hand, that PN → P = 0. Then, some branch of the logarithm (which we will denote log∗ ) is continuous at P and log∗ PN → log∗ P as N → ∞. Suppose we inductively define integers n k so that N

(log(1 + z k ) + 2πi n k ) = log∗ PN .

k=1

Then since log∗ PN converges, N

(log(1 + z k ) + 2πi n k ) k=1

converges; therefore log(1 + z k ) + 2πi n k → 0 as k → ∞. Since z k → 0 and log denotes the principal branch, it follows that n k = 0 for k sufficiently large. Hence ∞ k=1 log(1 + z k ) converges. 17.3 Proposition 3∞ If ∞ k=1 |z k | converges, k=1 (1 + z k ) converges.

17.1 Infinite Products

Proof Assume k>N

∞

k=1

243

|z k | converges and take N such that for k > N, |z k | < 12 . Then, for

z k3 z k2 1 1 | log(1 + z k )| = |z k − + − + · · · | ≤ |z k | 1 + + + · · · ≤ 2|z k |. 2 3 2 4 ∞ Hence k=N+1 log(1 + z k ) is convergent and by the previous proposition so is 3∞ k=1 (1 + z k ). 17.4 Definition 3∞ k=1 (1 + z k ) is called absolutely convergent if ∞ "

(1 + |z k |) converges.

k=1

17.5 Proposition An absolutely convergent product is convergent. Proof

3 According to Note (2) (following Definition of ∞ k=1 (1+|z k |) ∞ 17.1), the convergence 3∞ is equivalent to the convergence of |z |. Hence if (1 + |z |) converges k k k=1 k=1 3∞ so does ∞ |z | and by the previous proposition, so does (1 + z k ). k k=1 k=1 We wish to consider analytic functions defined by infinite products; i.e., functions of the form ∞ " f (z) = (1 + u k (z)). k=1

Recall that f is analytic if each function u k , k = 1, 2, . . . is analytic and the partial products converge to their limit function uniformly on compacta (Theorem 7.6). 17.6 Theorem

Suppose that u k (z) is analytic in a region D for k = 1,2, …, and that ∞ k=1 |u k (z)| 3∞ converges uniformly on compacta. Then the product k=1 (1 + u k (z)) converges uniformly on compacta and represents an analytic function in D. Proof

Let A be a compact subset of D. Since ∞ k=1 |u k (z)| converges uniformly on A, for sufficiently large k, |u k (z)| < 1 there. Hence, we may assume that 1 + u k = 0 for all k. If we then take N large enough so that ∞ k=N+1 |u k (z)| < /2, it follows, as in

244


the proof of Proposition 17.3, that

∞ log(1 + u k (z)) ≤ throughout A. k=N+1 That is, ∞ k=1 log(1 + u k (z)) converges uniformly on A to a limit function S(z). It follows that S(A) is bounded. Finally, since the exponential function is uniformly continuous in any bounded domain, N

PN (z) = exp log(1 + u k (z)) k=1

converges uniformly to its limit function e S(z).

E XAMPLES 3 k 1. ∞ k=1 (1 + z ) converges uniformly on any compact subset of the unit disc since any compact subset is contained in a disc of radius δ < 1. Hence ∞

|z k | ≤

k=1

and, by the M-test,

∞

k=1

2.

∞

δk =

k=1

δ 1−δ

|z k | is uniformly convergent. ∞ " 1 1+ z k

k=1

represents an analytic function in the half-plane D : Re z > 1. In any compact subset of D, Re z ≥ 1 + δ throughout so that 1 = 1 ≤ 1 , k = 1, 2, . . . . k z k Re z k 1+δ Hence

∞

1 kz k=1

and, consequently,

∞ " 1 1+ z k

k=1

are uniformly convergent.

♦

The Weierstrass Product Theorem. According to the Uniqueness Theorem (6.9), a nontrivial entire function cannot have an accumulation point of zeroes. That is, if {λk } → λ and if f is an entire function with zeroes at all the points λk , then f ≡ 0. On the other hand, an entire function may be zero at all the points of a sequence which


245

diverges to ∞. For example sin z is zero at all integral multiples of π. Similarly, e z −1 is zero at all the integral multiples of 2πi . The Weierstrass Product Theorem shows that these examples are in no way exceptional. 17.7 Theorem (Weierstrass) Suppose {λk }∞ k=1 → ∞. Then there exists an entire function f such that f (z) = 0 if and only if z = λk , k = 1, 2, . . .. Note: To define an entire function with zeroes at the points λk , it would seem natural to write ∞ " f (z) = (z − λk ). k=1

However, since λk → ∞, the terms of the product would not approach 1 (for fixed z) and hence the product would diverge. Instead, we consider the infinite product of linear functions given by ∞ " z f (z) = 1− , λk k=1 ∞ assuming for now that λk = 0. Indeed, if ∞ k=1 |1/λk | converges, k=1 |z/λk | converges uniformly on every compact set so that the product is uniformly convergent on compacta and gives the desired entire function. Moreover, if ∞ k=1 1/|λk | diverges ∞ but k=1 1/|λk |2 converges, we can modify the above construction by considering ∞ " z z/λk 1− e . f (z) = λk k=1

With the “convergence factors” e z/λk , the product is uniformly convergent on compacta since, for |λk | > 2|z|,

2 3 z z z z z log 1 − e z/λk = − − 2 − 3 + · · · + λk λk λ 2λk 3λk k 2 z2 1 1 1 z ≤ 2 + + ··· = 2. λk 2 4 8 λk Hence the series

∞

) * log (1 − z/λk )e z/λk , z = λk

k=1

is uniformly convergent and the uniformly convergent on compacta. product is m+1 By the same reasoning, if ∞ 1/|λ | converges for some positive integer k k=1 m and we consider the convergence factors E k (z) = exp z/λk + z 2 /2λ2k + · · · + z m /mλm k ,

246


it follows that the infinite product ∞ " z 1− E k (z) λk

k=1

is uniformly convergent on compacta and represents an entire function with the desired There are sequences {λk }, however, such that λk → ∞ and yet ∞ zeroes. N diverges for all N. (For example, {λ } = {log k}∞ .) Hence, for the 1/|λ | k k k=1 k=2 general case we must introduce a slight variation. Proof Assume for the moment that λk = 0 and set

z2 zk z E k (z) = exp + 2 + ···+ k . λk 2λk kλk Suppose, moreover, that |z| < M. Then since λk → ∞, for sufficiently large k, |λk | > 2|z| and ∞ j k

z 1 z z log 1 − E k (z) ≤ j ≤ ≤ k . jλ λk λk 2 j =k+1

k

Hence both ∞

k=1

∞ " z z E k (z) and 1− E k (z) log 1 − λk λk k=1

are uniformly convergent on compacta. Note also that the individual factors are zero only at the points λk , and by the definition of convergence the infinite product is zero at those points only. Finally, if we seek an entire function with zeroes at the origin as well, we need only set ∞ " z f (z) = z 1− E k (z) . λk P

k=1

E XAMPLES 1. To find an entire function f with a single ∞ zero at2every negative integer λk = −k, note that ∞ k=1 1/|λk | diverges but k=1 1/|λk | converges so that we can define ∞ " z −z/ k e 1+ . f (z) = k k=1


247

2. An entire function with zeroes at all the points λk = log k, k = 1, 2, . . ., is given by

∞ " z z2 zk z 1− exp + f (z) = z + ···+ . log k log k 2 log2 k k logk k k=2

3. An entire function with a single zero at every integer is given by

∞ ' ∞ " " z z/ k z −z/ k ( z2 1− 2 . e e 1− 1+ =z f (z) = z k k k k=1

♦

k=1

17.8 Proposition Let f (z) = z

∞ " k=1

z2 1− 2 k

.

Then f (z) = (sin πz)/π. Proof Consider the quotient Q(z) = z

∞ " k=1

z2 1− 2 k

4

sin πz.

Q is entire and zero-free. To show that Q is constant we seek estimates on its growth for large z. Assume then that 12 N ≤ |z| ≤ N. Then |Q(z)| is bounded by the maximum value assumed by Q on the square of side 2N + 1 centered at the origin (Theorem 6.13). We have already proved, however, (see Chapter 11.2) that along this square (which avoids the zeroes of sin πz), |1/ sin πz| ≤ 4. Moreover, ∞

N

∞ " z " z z 2 " z 2 1− 2 = 1− 2 1+ 1− k k k k k=1 k=1 k=N+1 ≤

N " k=1

e2|z/k|

∞ "

e|z

2/ k2 |

k=N+1

|z 2 | ≤ exp 2|z|(1 + log N) + N since

N

1 k=1

k

< 1 + log N and

∞

k=N+1

1 1 < . N k2

248


z –N – –¹ ²

– –¹ N ²

–N

–¹ N ²

N N + –¹ ²

√ Noting again that for large N , 2(1+log N) < N/2 ≤ |z|1/2 while |z 2 |/N ≤ |z|, it follows that 3 z ∞ 1 − z2 k=1 k 2 3/2 |Q(z)| = ≤ A exp(|z| ). sin πz By Theorem 16.12, then, we must have 3 z ∞ k=1 1 −

z2 k2

sin πz

= Ae Bz .

However, Q is an even function so that B = 0, and the constant A can be determined by noting that 1 z A = Q(0) = lim = . z→0 sin πz π Some consequences of the above proposition: i. Setting z =

1 2

, we have 1=

∞ 1 π " 1− 2 (2k)2 k=1

so that 2 = π

1·3 2·2

3·5 4·4

5·7 ··· 6·6

or π = 2·

2·2 1·3

4·4 3·5

6·6 ··· . 5·7

17.2 Analytic Functions Defined by Definite Integrals

249

ii. Suppose we expand the terms in the product to obtain an infinite series. Then we will have

∞ " z2 1− 2 sin πz = πz k k=1 ⎛ ⎤ ⎡ ⎞

∞

1

1 ⎠ z4 − + · · · ⎦ . z2 + 2 ⎝ = πz ⎣1 − k2 k2 j 2 k, j

k=1

A comparison with the familiar power series π 5z5 π 3z3 + − +··· 6 120

sin πz = πz − shows that

∞

π2 1 . = 6 k2 k=1

(See 11.2 for an earlier proof of this identity.)

17.2 Analytic Functions Defined by Definite Integrals We noted previously that Morera’s Theorem (7.4) can be used to prove the analyticity of certain functions given in integral form. We now examine this notion in somewhat greater detail. 17.9 Theorem Suppose ϕ(z, t) is a continuous function of t, a ≤ t ≤ b, for fixed z and an analytic function of z ∈ D for fixed t. Then

b

f (z) =

ϕ(z, t)dt

a

is analytic in D and f (z) =

b a

∂ (ϕ(z, t))dt. ∂z

(1)

Proof Since f is a continuous function of z, according to Morera’s Theorem (7.4), we need only prove that f (z)dz = 0 for any rectangle ⊂ D. We can reverse the order of integration, however, and write b b f (z)dz = ϕ(z, t)dt dz = ϕ(z, t)dz dt

a

a

250


since ϕ is continuous in t and in z. Thus, since ϕ is analytic in z,

b

f (z)dz =

0 dt = 0

a

We leave it as an exercise to show that f is given by the formula in (1).

E XAMPLES 1 1. f (z) = 0 dt/(t − z) is analytic in D = C\ [0, 1]. In fact, direct integration shows that f (z) = log(1 − 1/z), and we can use Theorem 10.8 to show that f is analytic in D. Recall then that Arg(1 − 1/z), as z traverses a closed curve, gives the number of zeroes minus the number of poles of 1 − 1/z that lie inside the curve. Yet if the curve is a simple closed curve encircling the interval [0, 1], because 1 − 1/z has one zero and one pole inside, Arg(1 − 1/z) = 0. The same argument shows that f has a jump discontinuity of 2πi as z crosses through any point x, 0 < x < 1 from the upper to the lower half-plane. ∞ 2. g(z) = 0 dt/(et − z) is analytic in C\ [1, ∞). Although g is given by an improper integral, it is the uniform limit of

N

g N (z) = 0

et

dt −z

on any compact subset of C\ [1, ∞), and hence g is analytic. As we shall see below, g has a “jump” of 2πi /x as z crosses from the upper half-plane to the lower half-plane through any point x > 1. ♦

17.10 Proposition Suppose that f and g are continuous real-valued functions on [a, b] and that f > 0 is also continuous. Then b g(t) F(z) = dt f (t) −z a is analytic outside the interval [α, β] where α = f (a), β = f (b) and lim [F(x + i y) − F(x − i y)] = 2πi

y→0+

g( f −1 (x)) for all x ∈ (α, β). f ( f −1 (x))

Proof The analyticity of F is proven in Theorem 17.9. By rationalizing the denominator, we obtain b b [ f (t) − x] g(t) g(t)dt F(x + i y) = dt + i y . 2 2 [ f (t) − x] + y [ f (t) − x]2 + y 2 a a

17.3 Analytic Functions Defined by Dirichlet Series

Hence

251

b

F(x + i y) − F(x − i y) = 2i y a

g(t)dt [ f (t) − x]2 + y 2

,

and setting t = f −1 (u), α = f (a), β = f (b) F(x + i y) − F(x − i y) = 2i

β α

yg( f −1 (u))du *. ) f ( f −1 (u)) (u − x)2 + y 2

We leave it as an exercise to complete the proof by showing that

β α

h(u)y du → πh(x) (u − x)2 + y 2

as y → 0 for any continuous function h on [α, β] and α < x < β.

17.3 Analytic Functions Defined by Dirichlet Series Series of the form

∞

an n=1

nz

are known as Dirichlet Series. Note that n −z = exp(−z log n) represents an entire function for every positive integer n. (log n is chosen as the principal value; i.e., log n is real-valued, so n −z is positive for all real z. The coefficients an , of course, can be any complex constants.) Since the partial sums are entire, a function f (z), defined by a Dirichlet series, is analytic in any region where the series converges uniformly. According to the theorems below, the natural regions of convergence for Dirichlet series are half-planes of the form Re z > x 0 , much as discs centered at the origin are the natural regions associated with power series. 17.11 Theorem an If ∞ n=1 z converges for z = z 0 , then it converges for all z in the half-plane H = n {z : Re z > Re z 0 }. Moreover, the convergence is uniform in any compact subset of H. Proof

an To show that ∞ n=1 z converges, we will show that the partial sums form a Cauchy n sequence. That is, we will show that N a a a N M n = + · · · nz Mz Nz n=M

is arbitrarily small for sufficiently large values of M.

252


Our proof is based on “summation by parts” and the following two observations: (i) Since

∞

an n z0 n=1

converges, there exists a positive constant A with T a n < A n z0

(2)

n=1

(ii)

for all positive integers T. 1 |w| 1 n w − (n + 1)w < n Re w+1 The above inequality follows easily from the usual M − L formula, since 1 1 − = w n (n + 1)w

n+1

wt −w−1 dt.

n

To complete the proof, let An =

n

ak , k z0 k=1

bn =

1 , with w = z − z 0 . nw

Then aN aM + · · · + z = (A M − A M−1 )b M + · · · + (A N − A N−1 )b N Mz N N−1

= −A M−1 b M + Ak (bk − bk+1 ) + A N b N .

(3)

k=M

−A M−1 b M and A N b N both go to zero for sufficiently large values of M and N, since |Ak | < A for all k, and |bn | = 1/n Re (z−z0 ) The remaining sum on the right side of (3) is also arbitrarily small for sufficiently large M since, according to (i) and (ii), it is bounded in absolute value by ∞

A|z − z 0 | , where δ = Re (z − z 0 ), k 1+δ

k=M

17.3 Analytic Functions Defined by Dirichlet Series

253

which is the “tail” of a convergent series. Hence ∞

an n=1

nz

converges. Finally, note that if K is a compact subset of H , there is a positive value of δ, with Re (z − z 0 ) > δ for all z in K , as well as a positive constant B with |z| < B throughout K . Hence the expression in (3) will have a uniformly small absolute value for all z in K , once M is sufficiently large. So the series converges to its limit function uniformly in K . Note that in the proof of Theorem 17.11, we never actually used the convergence of the Dirichlet series at z 0 . The only actual requirement for the conclusion was that there was a finite upper bound for the absolute value of its partial sums. E XAMPLE an Suppose an = (−1)n . Then ∞ n=1 z has bounded partial sums (although it diverges) n at z = 0. According to Theorem 17.11, then, it converges and represents an analytic function in the right half-plane: Re z > 0. The fact that it diverges at z = 0 also implies that its partial sums are not bounded for any value of z with a negative real part. ♦ 17.12 Theorem an If ∞ n=1 z converges for some, but not all, values of z, there exists a real constant n an x 0 (called the abscissa of convergence) such that ∞ n=1 z converges if Re z > x 0 n and diverges if Re z < x . 0

Proof Let x 0 be the greatest lower bound of the real parts of all the complex numbers z for an which ∞ n=1 z converges. By Theorem 17.11, if x 0 = −∞, the series converges n for all z. If the series neither converges for all z nor diverges for all z, −∞ < x 0 < ∞ and the theorem follows from Theorem 17.11 The abscissa of convergence of the Dirichlet series bears an obvious analogy to the radius of convergence of a power series. However, the analogy does not extend to the idea of absolute convergence. Power series converge absolutely in any compact subset of their region of convergence. On the other hand, consider the Dirichlet series ∞

(−1)n n=1

nz

As we mentioned above, the series converges (and represents an analytic function) in the right half-plane: Re z > 0. However, it converges absolutely only if Re z > 1.

254


This is the general situation with Dirichlet series. In addition to the half-plane of convergence H , there is a half-plane of absolute convergence H1, which may be a proper subset of H . 17.13 Theorem an Suppose ∞ n=1 z converges absolutely for some, but not all, values of z. Then there n an exists a constant x 1 (called the abscissa of absolute convergence) such that ∞ n=1 z n converges absolutely if Re z > x 1 and does not converge absolutely if Re z < x 1 . Proof a |a | n n z= x n n

an So if ∞ n=1 z is absolutely convergent at z 0 , it is also absolutely convergent at n all points z with Re z ≥ Re z 0 . The theorem follows with x 1 equal to the great an est lower bound of the real parts of all complex z for which ∞ n=1 z converges n absolutely. Note that if the coefficients an are all positive, the abscissas of convergence and absolute convergence must be identical. Otherwise there would be a real number x between them where the Dirichlet series is convergent but not absolutely convergent. But this is obviously impossible since the terms in the Dirichlet series, for real values of z, are all positive. E XAMPLE 1 The function ζ (z) is defined by the Dirichlet series ∞ n=1 z . This series converges n absolutely for Re z > 1, and diverges if Re z < 1. ♦ Since Dirichlet series converge uniformly within their half-plane of convergence, an they can be differentiated term-by-term. So if f (z) = ∞ n=1 z , then n ∞

−a log n n f (z) = . nz n=1

For any value of z within the half-planes of convergence for two Dirichlet series, we have : ∞ ∞ ∞

an bn an + bn + = . nz nz nz n=1

n=1

n=1

We can also multiply two Dirichlet series. Rewriting the product as another Dirichlet series involves a rearrangement of the terms, which is justified if the two series are absolutely convergent. Hence, within the half-planes of absolute convergence,

Exercises

255

we have

∞ ∞

an bn n=1

nz

n=1

nz

=

∞

cn nz n=1

with cn defined as the “convolution” of an and bn. That is,

ad bn/d cn = d|n

where the sum is taken over all the positive divisors of n. E XAMPLE ζ 2 (z) =

∞ ∞ ∞

1 1 d(n) = nz nz nz n=1

n=1

n=1

where d(n) equals the number of positive divisors of n.

Exercises 1. Prove

∞ " 1 1− 2 k

k=2

converges by finding an explicit formula for PN . 2. As above, prove ∞ " (−1)k 1+ k k=2

converges.

" i i ) diverges, but |1 + | converges. n n n n ∞ 3∞ 2 4. Show that if ∞ k=1 z k converges and k=1 |z k | converges, then k=1 (1 + z k ) converges. 5. Show that ∞ " (−1)k 1+ √ k k=2

3.* Prove that

"

(1 +

diverges even though

∞

(−1)k √ k k=2

converges. 3 2k 6. Prove that (1 +z)(1+z 2)(1+z 4) . . . = ∞ k=0 (1+z ) converges uniformly on compacta to 1/(1−z) in |z| < 1. [Hint: Find PN .] 7. Define an entire function g with single zeroes at and only at all the “squares” λk = k 2 ; k = 1, 2, . . .. √ √ 8. Show that one solution to (7) is given by sin π z/π z. 9. Prove that ∞ " 4z 2 1− . cos πz = (2k + 1)2 k=0

256


10. a. Define a function f , analytic in |z| < 1 and such that f (z) = 0

if and only if z = 1 −

1 ; k = 1, 2, . . . . k

[Hint: Find an entire function g with zeroes at λk = k, k = 1, 2, . . . and consider f (z) = g(1/(1 − z)).] b. Generalize the above results. 11. Given F(z) = ab ϕ(z, t)dt. Derive the formula for F (z) by writing F (z) =

b 1 1 F(ζ ) ϕ(ζ, t) dζ = dt dζ 2π i C (ζ − z)2 2π i C a (ζ − z)2

and switching the order of integration. 12. Complete Proposition 17.10 by splitting x− x+ β h(u)ydu into + + . α (u − x)2 + y 2 α x− x+

β

. 13. Show that f (z) =

1

dt 1 − zt 0

is analytic outside [1, ∞]. Find the discontinuity of f as z “crosses” a point x > 1. 14.* a. Let φ(n) be the Euler totient function; i.e., the number of positive integers not exceeding n, which are relatively prime to n. Prove that ∞

φ(n) nz n=1

is absolutely convergent for Re z > 2. φ(n) b. It can be shown that d|n φ(d) = n, for all n ≥ 1 [Apostol, p.26]. Show that ζ (z) ∞ n=1 n z = ζ (z − 1), for Re z > 2.

Chapter 18

Analytic Continuation; The Gamma and Zeta Functions

Introduction Suppose we are given a function f which is analytic in a region D. We will say that f can be continued analytically to a region D1 that intersects D if there exists a function g, analytic in D1 and such that g = f throughout D1 ∩ D. By the Uniqueness Theorem (6.9) any such continuation of f is uniquely determined. (It is possible, however, to have two analytic continuations g1 and g2 of a function f to regions D1 and D2 respectively with g1 = g2 throughout D1 ∩ D2 . See Exercise 1.) The Schwarz Reflection Principle (7.8) is an example of how, in some cases, an analytic function can be continued beyond its original domain of analyticity. In this chapter, we first examine the possibility of such “extensions” for functions given by power series. We then consider the classical Gamma and Zeta functions, defined originally by a definite integral and a Dirichlet series, respectively.

18.1 Power Series n As we have seen in Chapter 2, a power series, ∞ n=0 an z , may converge at some or all or even none of the points on its circle of convergence. As the examples below indicate, the convergence or divergence of the npower series at a point does not determine whether the function f (z) = ∞ n=0 an z , can or cannot be continued beyond that point. i.

∞

f (z) =

1 = z n for |z| < 1. 1−z n=0

Although the power series diverges at every point on the unit circle, f is analytic throughout the punctured plane z = 1. ∞ n /n 2 ) converges at all points on the unit circle; however, g(z) cannot be ii. (z n=1 continued analytically to a domain including z = 1 since g (z) =

∞

(n + 1)z n n=0

n+2

→ ∞ as z → 1− .

257

258

18 Analytic Continuation; The Gamma and Zeta Functions

18.1 Definition Suppose that f is analytic in a disc D and that z 0 ∈ ∂ D. Then f is said to be regular at z 0 if f can be continued analytically to a region D1 with z 0 ∈ D1 . Otherwise, f is said to have a singularity at z 0 . 18.2 Theorem ∞ n n If ∞ n=0 an z has a positive radius of convergence R, f (z) = n=0 an z has at least one singularity on the circle |z| = R. Proof If f were regular at every point on the circle of convergence, then for each z with |z| = R, there would exist some maximal z such that f could be continued to a region containing D(z; z ). Clearly z would depend continuously on z so that, since the circle |z| = R is compact, min z = > 0.

|z|=R

Hence, a function g would exist, analytic in D(0; R + ) and such that g = f in ∞ n D(0; R). But then g must have a power series representation n=0 bn z convergent ∞ n for |z| < R +. Yet since g(z) = f (z) = n=0 an z for |z| < R, by the Uniqueness Theorem for Power Series (2.12), an ≡ bn . Thus the radius of convergence would be R, and we have arrived at a contradiction. In general, it is difficult to determine when a function has a singularity at a particular point on the circle of convergence of its power series. The following theorem is one of the few results we have in this direction. 18.3 Theorem Suppose that ∞ an z n has a radius of convergence R < ∞ and that an ≥ 0 for n=0 n all n. Then f (z) = ∞ n=0 an z has a singularity at z = R. Proof By Theorem 18.2, f has a singularity at some point Reiα . If we consider the power series for f about a point ρeiα , with 0 < ρ < R: f (z) =

∞

n=0

bn (z − ρeiα )n =

∞

f (n) (ρeiα ) (z − ρeiα )n n! n=0

we see that the radius of convergence of this series is R − ρ. (If it were larger, the power series would define an analytic extension of f beyond Reiα ). Note, however,

18.1 Power Series

259

that for any non-negative integer j , f ( j )(ρeiα ) =

∞

n(n − 1) . . . (n − j + 1)an (ρeiα )n− j

n= j

so that, since an ≥ 0,

| f ( j ) (ρeiα )| ≤ f ( j ) (ρ).

Hence the power series expansion of f about ρ, ∞

f (n) (ρ) (z − ρ)n , n! n=0

must have radius of convergence R − ρ. On the other hand, if f were regular at z = R, the above power series would converge in a disc of radius greater than R − ρ; therefore, f is singular at z = R.

18.4 Definition n If f (z) = ∞ n=0 an z has a singularity at every point on its circle of convergence, then that circle is called a natural boundary of f . E XAMPLE

∞

k

z2 = z + z2 + z4 + z8 + · · ·

k=0

has radius of convergence 1. Yet as z → z 0 , where z 0 is any 2n th root of unity, all n the terms of the power series past z 2 approach 1, so that f (z) → ∞. Hence f is n singular at every 2 th root of unity, n ≥ 1. Since these are dense on the unit circle, that circle is a natural boundary for the power series. ♦ 2k k Similarly, if we set g(z) = ∞ k=0 (z /2 ) it is clear that g has the unit circle as a k natural boundary since g (z) = (1/z) ∞ z 2 → ∞ as z approaches any 2n th root ∞ 2k k 2 k=0 of unity. If we set h(z) = k=0 (z /2 ) then, while h has radius of convergence 1, all of its derivatives are bounded throughout |z| < 1. Nevertheless, according to the following theorem, h too has a natural boundary on the unit circle. 18.5 Theorem Suppose f (z) =

∞

k=0

ck z nk wi th

n k+1 > 1. k→∞ n k lim

Then the circle of convergence of the power series is a natural boundary for f .

260


Proof Since the result is independent of ck , we may assume without loss of generality that the radius of convergence is 1. Also, neglecting finitely many terms if necessary, we will assume that for some δ > 0 and for all k, n k+1 /n k > 1 + δ. Finally, it suffices to show that f is singular at the point z = 1. For the same result, applied to the series ∞ −iθ )n k shows that f is singular at any point z = e iθ . k=0 ck (ze Choose an integer m > 0 such that (m + 1)/m < 1 + δ and consider the power series g(w) obtained by setting z= and expanding the terms

wm + wm+1 2

wm + wm+1 2

n k

in the power series of f :

c0 wmn0 c0 n 0 wmn0 +1 c0 wm + wm+1 = + + · · · + n wmn0 +n0 g(w) = f n 2 2 0 2n 0 2 0 c1 c1 n 1 c1 + n wmn1 + n wmn1 +1 + · · · + n wmn1 +n1 2 1 2 1 2 1 + ··· . Note that in this expression no two terms involve the same power of w, since mn k+1 > mn k + n k holds whenever

n k+1 m+1 . > nk m

If |w| < 1, then

|w|m + |w|m+1 < 1, 2 and since f (z) is absolutely convergent for |z| < 1,

n k ∞

|w|m + |w|m+1 |ck | converges. 2 k=0

Hence for |w| < 1, g(w) is absolutely convergent. On the other hand, if we take w real and greater than 1, then wm + wm+1 >1 2 so that

n k ∞

wm + wm+1 ck 2 k=0

18.1 Power Series

261

diverges. Note, though, that the j th partial sums s j of the above series are exactly the n j (m + 1)-st partial sums of the power series for g. Hence the series for g(w) diverges and g, too, has radius of convergence 1. According to Theorem 18.2, g must have a singularity at some point w0 with |w0 | = 1. If w0 = 1, then wm + wm+1 0 0 0.

18.2 Analytic Continuation of Dirichlet Series

263

(The constants c p are determined in terms of the function which we will study in the next section. See Exercise 5.) E XAMPLE Let f (z) =

∞

n=0

Then d f (z) = z dz

n2 n z . +1

n2

∞

nz n . n2 + 1 n=0

Using one of the above formulae ∞

n=0

∞ ∞

n n −nt n z = (e cos t)z dt n2 + 1 0 n=0 ∞ t e cos t dt for |z| < 1. = et − z 0

Thus

∞

f (z) = z 0

et cos t dt. (et − z)2

[Alternatively, we could write ∞ ∞

1 1 1 f (z) = zn = − z n , etc. . . . .] 1− 2 2 n +1 1−z n +1 n=0

♦

n=0

18.2 Analytic Continuation of Dirichlet Series Dirichlet series, unlike power series, do not necessarily have a singularity on their boundary of convergence. For example, we will see in the next section that ∞

(−1)n n=1

nz

can actually be continued to the full complex plane. However, if all the coefficients an are positive, we have the following analogue of Theorem 18.3. 18.6 Landau’s Theorem Suppose that an ≥ 0 for all n, and that b is the real boundary point of the region of convergence of ∞

an f (z) = . nz n=1

Then f has a singularity at b.

264


Proof We will show that if f is regular at b; that is, if it can be analytically extended to a region containing the point b, then the Dirichlet series will converge at some real number less than b, which contradicts the definition of b. Toward that end, choose a real number a > b, and consider the power series representation of f , centered at z = a. Since ∞

an (− log n)k f (k) (z) = , nz n=1

the power series representation for f in a disc centered at z = a is f (z) =

∞

ck (z − a)k with ck =

k=0

∞

an (− log n)k n=1

n a k!

(1)

If f is regular at z = b, the radius of convergence of the series in (1) is greater than a − b so that the series converges at a point of the form b − ε, with ε > 0. That is, ∞

∞

an (log n)k )(a − b + ε)k ( n a k!

(2)

k=0 n=1

converges. Since an ≥ 0 for all n, all the terms in (2) are nonnegative. Hence it is an absolutely convergent series and, as such, its terms can be rearranged in any form. Suppose then that we first sum over k. Then ∞

(log n)k k=0

k!

)(a − b + ε)k = e(a−b+ε) log n = n a−b+ε

and the convergent series in (2) becomes ∞

an n a−b+ε n=1

na

which is exactly the Dirichlet series with z = b − ε.

18.7 Corollary If a Dirichlet series has nonnegative coefficients and can be analytically continued to the entire complex plane, then it converges throughout the complex plane. Proof If the series did not converge for all z, according to Theorem 18.6, the function represented by the Dirichlet series would have a singularity at the real boundary point of its region of convergence.

18.3 The Gamma and Zeta Functions

265

18.3 The Gamma and Zeta Functions The Gamma Function. Consider the integral ∞ In = e−t t n dt n = 0, 1, 2 . . . . 0

Integration by parts shows that

∞

In = n

e−t t n−1 dt = n In−1 .

0

Since I0 = 1, the above recurrence relation implies In = n! for all positive integers n. Moreover, the above integral allows us to extend this “factorial” function to the complex plane. Note that |t z | = |e z log t | = e(Re z) log t = t Re z for t ≥ 0 so if we replace n by the complex variable z, the resulting function f (z) = ∞that −t t z dt is uniformly convergent for Re z > −1. A translate of this function, e 0 ∞ e−t t z−1 dt, (1) (z) = 0

is the classical Gamma Function. Thus is analytic in the right half-plane Re z > 0 and (n) = (n − 1)! for all positive integers n. It is clear that has a singularity at z = 0 since ∞ −t e () = dt → ∞ as → 0+ . 1− t 0 On the other hand, although (1) defines only in the right half-plane, the function can be extended to the whole plane with the exception of isolated poles. We may carry out this extension in several ways. I. Integration by parts shows that (z + 1) = z(z) for Re z > 0, or equivalently,

(z + 1) for Re z > 0. (2) z Identity (2) allows us to define an extension of to the half-plane Re z > −1, z = 0. This extension is analytic for −1 < Re z < 0 and is continuous along the nonzero y-axis since the “original” is continuous on the line Re z = 1. That is, (z) =

lim (z) = lim

z→iy

z→iy

(i y + 1) (z + 1) = = (i y), z iy

y = 0.

266


Hence by Morera’s Theorem the extended function is analytic throughout Re z > −1, z = 0. Identity (2) also reveals the nature of the singularity at z = 0, since as z → 0 (z) =

(z + 1) (1) 1 ∼ = . z z z

Hence has a simple pole with residue 1 at z = 0. Continuing in the same manner, we can define (z + 1) (z + 2) = for Re z > −2, z z(z + 1) (z + 3) (z) = for Re z > −3, . . . , z(z + 1)(z + 2) (z + k + 1) (z) = for Re z > −k − 1. z(z + 1) · · · (z + k)

(z) =

(3)

Note then that the only singularities are the isolated (simple) poles at the non-positive integers, and as z → −k (z) ∼

(−1)k (1) = . (−k)(−k + 1) · · · (−1)(z + k) k!(z + k)

Hence Res((z); −k) =

(−1)k . k!

II. Set (z) = 1 (z) + 2 (z), where

1

1 (z) =

0

2 (z) =

e−t t z−1 dt

∞

e−t t z−1 dt, Re z > 0.

1

Since |t z−1 | = t Re z−1 , 2 is uniformly convergent for all z and represents an entire function. Thus, to extend , we need only to extend 1 . But for Re z > 0

1 t2 1−t + − + · · · t z−1 dt 1 (z) = 2! 0 1 1 1 z+1 t − +··· = t z−1 dt − t z dt + 2! 0 0 0 1 1 1 = − + − +··· . z (z + 1) 2!(z + 2) The above series defines an analytic extension of 1 to the whole plane except for isolated poles at 0, −1, −2, . . .. Note again that Res(; −k) = Res(1 ; −k) =

(−1)k . k!


267

III. Using the fact that (1 − t/n)n converges to e−t as n → ∞, one can show that n t n (z) = lim t z−1 1 − dt n→∞ 0 n n 1 = lim n t z−1 (n − t)n dt, Re z > 0. n→∞ n 0 (See Exercise 7.) Integrating by parts, we have

1 n n z · t (n − t)n−1 dt n→∞ n n z 0 n 1 n(n − 1) · · · 1 = lim n t z+n−1 dt n→∞ n z(z + 1) · · · (z + n − 1) 0 nz 1 2 n ··· . = lim n→∞ z z+1 z+2 z+n

(z) = lim

Thus,

z 1 z = lim zn −z (1 + z) 1 + ··· 1 + (z) n→∞ 2 n n " z = lim zn −z . 1+ n→∞ k k=1

To examine the above limit, we insert “convergence factors” e−z/ k and obtain n " z −z/k 1 = lim zn −z e z(1+1/2+···+1/n) e 1+ (z) n→∞ k k=1 n " z −z/ k z(1+1/2+···+1/n−log n) z . e 1+ = lim e n→∞ k k=1

By the lemma below, 1 + 12 + · · · + 1/n − log n approaches a positive limit γ (known as the Euler constant) so that ∞ " z −z/ k 1 = zeγ z e 1+ . (z) k k=1

Using the above identity to define an extension of to the left half-plane, we obtain

∞ 2 " z sin πz 1 1 − 2 = −z = −z 2 . (z)(−z) π k k=1

Thus

(z)(−z) =

−π , z sin πz

268


and since (1 − z) = −z(−z), (z)(1 − z) =

π . sin πz

(4)

Two immediate consequences of identity (4) are i. is zero-free, √ ii. ( 12 ) = π. Applying the identity (z + 1) = z(z), we have also (3/2) = √ √ 1 2 π, (5/2) = 3 π/4, etc. 18.8 Lemma If sn = 1 + 12 + · · · + 1/n − log n, then limn→∞ sn exists. This limit is called the Euler constant, γ . Proof tn = 1 + 12 + · · · + 1/(n − 1) − log n increases with n. Geometrically this is obvious since tn represents the n area of the n − 1 regions between the upper Riemann sum and the exact value for 1 (1/x) d x. We can write tn =

n−1

1 k=1

and lim tn =

n→∞

k

− log

k+1 k

1 − log 1 + . k k

∞

1 k=1

The series above converges to a positive constant since 1 1 1 1 1 1 0 < − log 1 + = 2 − 3 + 4 − +··· ≤ 2. k k 2k 3k 4k 2k This proves the lemma, because limn→∞ sn = limn→∞ tn .

The Zeta Function. Recall that the Zeta Function ζ(z) is defined by the Dirichlet series 1 1 ζ (z) = 1 + z + z + · · · , Re z > 1. 2 3 This function is of special interest in number theory because it provides a link between the prime numbers and analytic function theory. To see this connection, note that 1 1 1 1 ζ (z) = z + z + z + · · · z 2 2 4 6 so that

1 1 1 1 − z ζ (z) = 1 + z + z + · · · . 2 3 5


269

Similarly, 1 1 1 1 1 1− z 1 − z ζ (z) = 1 + z + z + z + · · · , 2 3 5 7 11 and because of the unique prime factorization of the integers, we can continue indefinitely to obtain (in the limit) " 1 1 − z ζ (z) = 1. p p prime

!

That is ζ (z) = 1

" 1 1 − z , Re z > 1. p

(5)

p prime

To best exploit identity (5), we need to extend ζ beyond the domain Re z > 1. Note that ζ does have a singularity at z = 1 since ζ (1 + ) → ∞ as → 0+ . We shall see below that this is the only singularity of the ζ function. We extend ζ by the method of moments. Note that ∞ 1 ∞ −t z−1 (z) e−nt t z−1 dt = z e t dt = z n 0 n 0 so that ∞

∞ ∞

1 z−1 −nt dt (z) = t e nz 0 n=1 n=1 ∞ z−1 t dt. = et − 1 0 That is, ζ (z) =

1 (z)

0

∞

t z−1 dt et − 1

or 1 ζ (z) = (z)

1 0

t z−1 dt + et − 1

∞ 1

t z−1 dt . et − 1

(6)

Recall that 1/ ∞(z) (with the appropriate limiting value of zero at the poles of ) is entire, as is 1 (t z−1 /(et −1))dt. Furthermore, the Laurent Expansion for 1/(et −1) around t = 0, et

1 1 = + A0 + A1 t + A2 t 2 + · · · , −1 t

270


converges absolutely for t = 1 so that

1 0

t z−1 dt = et − 1

1

(t z−2 + A0 t z−1 + A1 t z + · · · )dt

0

A0 A1 1 + + + ··· (7) z−1 z z+1 1 provides an analytic extension of 0 (t z−1 /(et − 1)) dt except for isolated poles. According to (6), then A0 A1 1 1 + + · · · + g(z) (8) ζ (z) = (z) z−1 z z+1 =

where g(z) is entire. Note that while he bracketed expression above has a simple pole at z = 1 as well as at every non-positive integer, all these poles are cancelled by the zeros of 1/ (z) except z = 1. Hence ζ has a single (simple) pole at z = 1 with residue 1. For future reference, then, we record 18.9 Theorem The only singularity of the Zeta function ζ (z) is a simple pole with residue 1 at z = 1. According to (5), ζ is zero-free for Re z > 1. The celebrated Riemann hypothesis asserts that all the complex zeroes of the Zeta function lie on the line Re z = 12 . While this hypothesis has neither been proved nor disproved, the following theorem offers an important extension of the zero-free region of ζ. 18.10 Theorem ζ is zero-free throughout Re z ≥ 1. Proof A key element in the proof is the observation that if ζ(1 + i a) = 0, then the function f (z) = ζ (z)ζ (z + i a) is entire. At z = 1, the pole of ζ(z) is cancelled by the zero of ζ (z + i a). Also, since ζ (z) is real-valued for real z, according to the Schwarz Reflection Principle, ζ (z) = ζ (z). Hence ζ (1 − i a) = 0, and the pole of ζ(z + i a) at z = 1 − i a is cancelled by the zero of ζ (z) at that point. Note that f (z − i a) = ζ(z − i a)ζ (z) will also be entire as will the product g(z) = f (z) f (z − i a) = ζ 2 (z)ζ (z + i a)ζ (z − i a). The desired contradiction will be based, in part, on the fact that the Dirichlet series for g(z) has all nonnegative coefficients. To see that, we first consider log(g(z)) which, according to Euler’s formua for ζ(z), is given by

log(g(z)) = [ − 2 log(1 − p −z ) − log(1 − p −z+ia ) − log(1 − p −z−ia )] p

1 (2 + p −ina + p ina ) = nz np p,n

Exercises

271

The sum is taken over all primes p and all positive integers n. Since 2 + p−ina + pina = 2 + 2 cos(na log p) ≥ 0, all of the coefficients in the above Dirichlet series for log(g(z)) are nonnegative. But if a Dirichlet series S(z) = an n −z has all nonnegative coefficients, so does e S(z) =

" ak n kz k! n n k

Hence g(z) represents an entire function whose Dirichlet series has all nonnegative coefficients. According to Corollary 17.12, then, its Dirichlet series must converge for all z! But this is clearly impossible. Since all of the coefficients of g(z) =

2 n

−z

n

n −z−ia

n

n −z+ia

n

are nonnegative, the sum is clearly positive for all real z. Moreover, the sum must be larger than the sum over any subset of the positive integers. So if we consider nonnegative real values of z and limit ourselves to the subseries corresponding to integers n of the form 2k , we have |g(z)| >

1 1 1 · · −z 2 −z−ia (1 − 2 ) 1 − 2 1 − 2−z+ia

Finally, since z is nonnegative, |(1 − 2−z−ia )(1 − 2−z+ia )| ≤ 4, and |g(z)| >

1 4(1 − 2−z )2

Letting z → 0 through positive real values, then, shows that the Dirichlet series for g diverges at 0.

Exercises 1. Let f (z) = log z, Re z > 0, Im z > 0. Let g1 be the continuation of f to the plane minus the negative axis (and 0) and let g2 be the continuation of f to the plane minus the negative imaginary axis (and 0). Show that g1 = g2 throughout the third quadrant. 2.* a. Suppose f (z) = an z n has radius of convergence 1 and assume that an analytic continuation of f has a pole at z = 1. Show that an z n diverges at every point on the unit circle. (Hint: as z → 1 from below, along the x−axis.) Show that if {an } → 0, then (1 − z) f (z) → 0 b. Generalize the result; i.e. show that if f (z) = an z n has a positive radius of convergence and n has a pole at any point on its circle of convergence, then z an analytic continuation of a n an z n diverges at all points on the circle of convergence. n n 3. Prove: If ∞ n=0 (−1) an z , an ≥ 0 has a finite radius of convergence, then it has a singularity on the negative axis.

272


4. Define an analytic continuation of a.

∞

∞

5. Show that

zn

√ , n=1 3 n

0

e−nt t p−1 dt =

6. Use the Gamma Function to show

∞ 0

7. Prove

2

∞

zn

n=0 n 2 + 1

.

( p) for p > 0. np

e−t dt =

√ π . 2

t n t z−1 1 − dt, Re z > 0. n→∞ 0 n

(z) = lim [Hint: First show, for t ≤ n,

b.

n

t2 t 0 ≤ e−t/n − 1 − ≤ 2 n 2n

and then use the identity a n − bn ≤ na n−1 (a − b) for a > b to show

2 −t e − 1 − t ≤ e−t et ·] n 2n

8.* Use the product formula for 1/ (z) to prove that (1) = −γ . 9. Show that

1 1 1 1 − z + z − z + −··· 2 3 4

can be continued analytically to the full plane. That is, show that it represents an entire function. 10. Use identity (5) to prove p prime (1/ p) diverges.

Chapter 19

Applications to Other Areas of Mathematics

Introduction We have already seen, especially in Chapter 11, how the methods of complex analysis can be applied to the solution of problems from other area of mathematics. In this chapter we will get some insight into the fantastic breadth of such applications. For that reason, the topics chosen are rather disparate. Section 19.1 involves calculating the total variation of a real function, and illustraties how the methods of Chapter 11 can be applied to yet another nontypical problem. Section 19.2 offers a proof of the classic Fourier Uniqueness Theorem using two preliminary results from real analysis and a surprising application of Liouville’s theorem. In Section 19.3 we see how the use of a generating function allows complex analytic results to be applied to an infinite system of (real) equations. Generating functions are also the key to the four different problems in number theory that comprise section 19.4. Finally, in section 19.5, we offer a well-trimmed analytic proof of the prime number theorem based on properties of the Zeta function and another Dirichlet series.

19.1 A Variation Problem P ROBLEM Calculate the total variation of sin2 x/x 2 on (−∞, ∞). Note: (sin2 x/x 2 ) is nonnegative and clearly has one local maximum between each pair of adjacent zeros. Hence the total variation is simply twice the sum of its maximum values. This problem is related to the type of sums encountered in 11.2. The novelty here lies in the fact that while we will not explicitly determine the maximum points x k ; k = 1, 2, . . . , we will be able to find the desired sum: ∞

sin2 x k k=1

x k2

.

273

274

19 Applications to Other Areas of Mathematics

S OLUTION The maximum points of sin2 x/x 2 are given by those zeros of its derivative that are not zeroes of sin x/x, and these are the (real) solutions of x cos x − sin x = 0, or simply tan x = x. We leave it as an exercise to verify that tan z = z has no nonreal solutions. Thus we need only sum the values of sin2 x/x 2 at the zeroes of tan z − z. Except for z = 0, all the zeroes of tan z − z are simple. We recall that f / f has residue 1 at every simple zero of f , thus we see that

sin2 x k x k2

=

1 2πi

f 1 (z)dz − Res( f 1 ; 0) CN

where

sin2 z tan2 z z 2 (tan z − z) and where the sum is taken over all the nonzero maximum points x k that lie inside C N . However, we must find a sequence of suitable contours C N which contain all the points x k (in the limit) and are such that lim N→∞ C N f 1 (z)dz can be determined. If we take C N to be the square of side 2πN centered at z = 0, it follows that | tan z| < 2 throughout C N . [See 11.2 and use the fact that tan z = cot(π/2 − z)]. However, there are several difficulties. Not only is sin z unbounded throughout C N , but f 1 (z) introduces an infinity of residues at the poles of tan z. To overcome these difficulties we replace sin2 z tan2 z z2 by another analytic function with the same values at the zeroes of tan z − z. Thus we substitute z 2 for tan2 z and since f 1 (z) =

sin2 z = we consider f 2 (z) =

tan2 z , 1 + tan2 z

z2 . (1 + z 2 )(tan z − z)

Again, however, there is a difficulty. While f 2 (z) → 0 along C N (as N → ∞), it is not true that C N f 2 (z)dz → 0 since for large z, f 2 (z) ∼ −1/z. (See Exercise 2.) Hence we make one more adjustment and consider finally f 3 (z) =

z tan z . (1 + z 2 )(tan z − z)

Note that f 3 is analytic at the poles of tan z and that | f 3 (z)| ≤ A/|z|2 throughout C N . Thus it follows that f 3 (z)dz → 0 as N → ∞, (1) CN

19.2 The Fourier Uniqueness Theorem

275

while, on the other hand,

f 3 (z)dz → 2πi Res( f 3 (z)) CN ⎡ ⎤ ∞ 2

sin x k ⎢ ⎥ = 2πi ⎣ + Res( f 3 ; 0) + Res( f 3 ; i ) + Res( f 3 ; −i )⎦ . 2 xk k=1

(2)

x k =0

A direct calculation shows that Res( f 3 ; i ) = Res( f 3 ; −i ) = (1 − e2 )/4. Expanding z sin z (1 + z 2 )(sin z − z cos z)

f 3 (z) =

around z = 0, we see that Res( f 3 ; 0) = 3. Since sin2 x/x 2 = 1 at x = 0, a comparison of (1) and (2) shows that

∞

sin2 x k sin2 x = 2 = e2 − 5. ♦ Var x2 x k2 k=1

19.2 The Fourier Uniqueness Theorem Suppose f is Lebesgue-integrable on (−∞, ∞). Then, by definition, ∞ ∞ | f (t)|dt < ∞, and for any real x, −∞ f (t)ei xt dt exists. The function fˆ(x) −∞ ∞ defined as fˆ(x) = −∞ f (t)ei xt dt is called the Fourier transform of f . The question we consider is whether f is uniquely determined by fˆ. That is, does fˆ ≡ 0 imply f ≡ 0 in the L 1 sense: that is, almost everywhere? The answer is yes and is usually found by appealing to an inversion formula which allows one to recover f from fˆ. The analytic proof below is somewhat more direct. We do, however, require two elementary results from the Lebesgue theory. 19.1 Lemma If gn (x) is a sequence of measurable functions such that |gn (x)| ≤ G(x), where G(x) is integrable, for all values of x and n, and if lim gn (x) = g(x)

n→∞

for all x, then

lim

∞

n→∞ −∞

gn (x) d x =

∞

−∞

g(x) d x.

276


Proof

See [Titchmarsh, p. 345]. 19.2 Lemma If f is integrable and if

a

−∞

f (t)dt = 0 for all real a, then f = 0 almost everywhere.

Proof

[Titchmarsh, p. 360]. 19.3 Fourier Uniqueness Theorem If f is integrable and if fˆ(x) =

∞ −∞

f (t)ei xt dt ≡ 0,

then f = 0 almost everywhere. Proof By hypothesis, for any real a, a f (t)ei x(t −a)dt = − −∞

If we set

L(z) =

a

−∞

∞

f (t)ei x(t −a) dt.

(3)

a

f (t)eiz(t −a) dt,

R(z) = −

∞

f (t)eiz(t −a) dt,

a

then L(z) is defined for Im z ≤ 0 while R(z) is defined for Im z ≥ 0. Moreover each is analytic in the interior of its domain, continuous to the boundary (according ∞ to Lemma 19.1), and bounded by −∞ | f (t)|dt. Since, by (3), they agree on the boundary, we can invoke Theorem 7.7 to prove that the “combined” function L(z) Im z ≤ 0 F(z) = R(z) Im z > 0 is entire. By Liouville’s Theorem (5.10), the boundedness of F implies that F is ∞ constant. Finally, setting z = Ni we have F(Ni) = R(Ni) = a − f (t)e−N(t −a) dt which approaches 0 as N → ∞, by Lemma 19.1. Hence F(z) ≡ 0. In particular, setting z = 0 yields F(0) =

a

−∞

f (t)dt = 0.

Since this holds for all real a, according to Lemma 19.2, f everywhere.

= 0 almost

19.3 An Infinite System of Equations

277

19.3 An Infinite System of Equations Consider the infinite system of equations a1 + b1 = 2 a2 + 2a1b1 + b2 = 4 a3 + 3a2 b1 + 3a1b2 + b3 = 8 ··· n n an−1 b + an−2 b2 + · · · + bn = 2n . an + 1 2 ··· P ROBLEM Assuming a1 = b1 = 1 and ak , bk ≥ 0 for all k, does there exist a solution to the system above other than an ≡ 1, bn ≡ 1? Note that if we do not insist ak , bk ≥ 0, there are an infinite number of solutions, since each equation introduces two new unknowns. Somewhat surprisingly, then, the answer to the given problem is no. (Somehow, any scheme to solve the equations successively is doomed to eventually introduce negative values for ak or bk .) S OLUTION Assume the sequences {an } and {bn } yield a solution to the above system. To study these sequences, we consider their “exponential generating functions”. That is, we define ∞ ∞

an z n bn z n f (z) = , g(z) = n! n! n=0

n=0

with a0 = b0 = 1. Since ak , bk ≥ 0, for all k, it follows that ak , bk ≤ 2k , so that both f (z) and g(z) are entire functions. Note that f (z)g(z) =

∞

Cn z

n

n=0

n

an− j b j where Cn = (n − j )! j ! j =0

=

n

n an− j b j j =0

so that according to the hypothesis f (z)g(z) =

∞ n n

2 z n=0

n!

= e2z .

j

n!

,

278


Thus f and g are entire functions with no zeros and both are of linear order. According to Theorem 16.13, then, f (z) = eαz+β and g(z) = eγ z+δ . Since f (0) = g(0) = a0 = b0 = 1, f (z) =

eαz

and g(z) =

eγz .

On expanding

α2 z 2 + ··· = 1 + z + 2! γ 2z2 + ··· = 1 + z + = 1 + γz + 2!

f (z) = eαz = 1 + αz + g(z) = eγz

a2 z 2 + ··· 2! b2 z 2 + ··· 2!

it follows that α = γ = 1 and hence an ≡ bn ≡ 1.

♦

19.4 Applications to Number Theory Like the example in Section 19.3, all of the problems in this section involve sequences of real numbers, and in each case we will consider an appropriate generating function. Note, therefore, that the ordinary generating function of a sequence {cn } is the power series cn z n . I A Partition Problem P ROBLEM Can the positive integers {1, 2, 3, . . .} be partitioned into a finite number of sets S1 , S2 , . . . , Sk , each of which is an arithmetic progression - that is, S1 = {a1 , a1 + d1 , a1 + 2d1, . . .} S2 = {a2 , a2 + d2 , a2 + 2d2, . . .} ··· Sk = {ak , ak + dk , ak + 2dk , . . .} and such that there are no equal common differences (i.e. di = d j for i = j )? Note that if we allowed equal common differences, the answer would be obviously yes. For example, we could take S1 = { odd integers } and S2 = { even integers }. With the given hypothesis, as we shall see, the answer is no. S OLUTION For any set S ={a, a + d, a + 2d, . . .}, consider the sequence {cn }, where cn = 1 if n ∈ S, and cn = 0 otherwise. Then the generating function for the sequence {cn } is

19.4 Applications to Number Theory

the geometric series

279

zn =

n∈S

za 1 − zd

Assume that S1 , S2 , . . . , Sk (as above) partition the set of positive integers. Taking the various generating functions, we would then have ∞

zn =

zn +

n∈S1

n=1

or

n∈S2

zn + · · · +

zn

n∈Sk

z a1 z a2 z ak z = + + · · · + . 1−z 1 − z d1 1 − z d2 1 − z dk

(4)

Since di = d j , we may assume d1 > d j for j = 1. It follows that as z → e2πi/d1 the first term on the right side of (4) will approach infinity while all the others approach a finite limit. This clearly contradicts (4) and hence no partition of the desired type is possible. II Making Change Suppose the coins in a certain country had the values 3 cents, 8 cents, and 15 cents. How many ways could we offer change for $1.00 using those coins? A direct check would show that the answer is 15, but would not really suggest an approach to the more general question: Can we find a formula for C(n), the number of ways that an arbitrary nonnegative integer n can be expressed as a (nonnegative) integral combination of 3’s, 8’s and 15’s ? Note that, by the familiar rule for the product of power series, the generating function for the sequence {C(n)} is ∞

C(n)z n = (1 + z 3 + z 6 + · · · )(1 + z 8 + z 16 + · · · )(1 + z 15 + z 30 + · · · )

n=0

or

∞

C(n)z n =

n=0

1 1 1 · · 1 − z 3 1 − z 8 1 − z 15

(5)

So our search for C(n) leads to a search for the coefficient of z n in the product on the right side of (5). To that end, we would like to find the partial fraction decomposition for the function R(z) =

1 1 1 · · 3 8 1 − z 1 − z 1 − z 15

Note that since R is a proper rational function; i.e., R(z) → 0 as z → ∞, the partial fraction decomposition is simply the sum of the principal parts of the Laurent expansions about each of the singularities (see Theorem 9.13). Let α = e2πi/3 , β = e2πi/8 , γ = e2πi/15

280


and express R(z) as 3 "

R(z) = −

n=1

8 15 1 " 1 " 1 . z − αn z − βn z − γn n=1

n=1

R has a pole of order three at z = 1(= α 3 = β 8 = γ 15 ), double poles at α(= γ 5 ) and at α 2 (= γ 10 ) and simple poles at its other 19 singularities. Hence R has a partial fraction decomposition of the form R(z) =

3

k=1

dj ak bk ck + + + k k 2 k (z − 1) (z − α) (z − z j ) (z − α ) 2

2

19

k=1

k=1

j =1

(6)

where the numbers z j represent the 19 simple poles of R(z). Note that if we can find all the coefficients ak , bk , ck and d j , we will be able to find the desired coefficient of z n in R(z). This is because repeated differentiation of the familiar identity 1 −1 −1 z z2 = = (1 + + 2 + · · · ) z−ω ω(1 − z/ω) ω ω ω shows that 1 k −1 n + 1 (n + 2)(n + 1) ) is n+1 , n+2 , and − z−ω ω ω 2ωn+3 for k = 1, 2, and 3, respectively. (7) the coefficient of z n in (

Hence we can can apply the techniques discussed in Chapter 10 to find all of the coefficients ak , bk , ck and d j , and we can then complete the job by applying formula (7) to each of the terms of (6), taking ω = 1 and k = 1, 2, and 3; ω = α and ω = α 2 with k = 1 and 2; and w = z j for each value of j (with k = 1). For obvious reasons, we will choose to skip the remaining details. Instead, we will leave the reader the challenge of completing a similar exercise with more "reasonable" numbers. However, we can easily answer a related question: That is, putting aside the problem of finding the exact formula for C(n), can we find an asymptotic formula? In other words, can we identify a familiar function f (n) with the property that C(n)/ f (n) → 1 as n → ∞ ? To find f (n), we need only look at formulas (6) and (7) to see that the "largest" contribution to C(n), the only one with order of magnitude n 2 , corresponds to ω = 1 and k = 3. (Note that |ω| = 1 in all cases). Thus an asymptotic formula for C(n) is −a3 n 2 . 2 In fact, we can calculate a3 as lim (z − 1)3 R(z) = − lim

z→1

z→1

1−z −1 1−z 1−z · · = 360 1 − z 3 1 − z 8 1 − z 15


281

Finally, then, we have the asymptotic formula: C(n) ∼

n2 720

The above example is easily generalized as follows: Suppose we are given any relatively prime set of integers S = {a1 , a2 , · · · , ak } and we would like to find C(n), the number of ways that an arbitrary nonnegative integer n can be expressed an a (nonnegative) integral combination of the elements of S. As in the earlier example, the generating function for C(n) is given by ∞

C(n)z n = (1 + z a1 + z 2a1 + · · · )(1 + z a2 + z 2a2 + · · · ) · · ·

n=0

× (1 + z ak + z 2ak + · · · ) 1 1 1 = · ··· . 1 − z a1 1 − z a2 1 − z ak Thus C(n) is the coefficient of z n in R(z) =

1 1 1 · ··· , a a 1 2 1−z 1−z 1 − z ak

and we can find C(n) by obtaining the partial fraction decomposition. Again, as in the special case treated above, finding an exact formula for C(n) is rather foreboding. However, to find an asymptotic formula for C(n), we need only establish the following points (whose proofs we leave to the reader): (i) R has a pole of order k at z = 1. (ii) Every other pole of R(z) is of order less than k. (iii) k n k−1 1 n+k −1 The coefficient of z n in is ∼ k−1 1−z (k − 1)! (iv) The coefficient a−k in the Laurent expansion of R(z) about the point z = 1 is (−1)k /(a1 a2 · · · ak ) Combining all of the above gives the general asymptotic formula: C(n) ∼

n k−1 (k − 1)!a1 a2 · · · ak

An immediate, but nontrivial, corollary of the above formula is the fact that for sufficiently large values of n, C(n) > 0. That is, every “sufficiently large” integer can be expressed as a nonnegative integral combination of the integers in S. This also highlights the necessity of the condition that S be a relatively prime set of integers.

282


Otherwise, all the elements of S would be multiples of some integer greater than 1, and so would any nonnegative integral combination of the elements of S. III An Identity of Euler’s Consider expressing n as the sum of distinct positive integers, i.e., where repeats are not allowed. (So for n = 6, we have the expression 1 + 2 + 3 as well as 1 + 5 and 2 + 4, and just plain 6.) Also consider expressing n as the sum of positive odd integers, but this time where repeats are allowed. (So for n = 6, we have 1 + 5, 3 + 3, 1 + 1 + 1 + 3, and 1 + 1 + 1 + 1 + 1 + 1.) In both cases we obtained four expressions for 6, and a theorem of Euler’s says that this is no coincidence. 19.4 Euler’s Theorem The number of ways of expressing n as the sum of distinct positive integers is the same as the number of ways of expressing n as the sum of (not necessarily distinct) odd positive integers. Proof To prove the theorem, we produce two generating functions. The latter is exactly the “coin-changing” function where the coins have denominations 1, 3, 5, 7, . . . This generating function is given by 1 (1 − z)(1 − z 3 )(1 − z 5 ) · · · Note that the infinite product in the denominator converges uniformly on compacta and represents a nonzero analytic function for |z| < 1. The other generating function is not of the coin-changing variety because of the distinctness condition. However this generating function is the product (1 + z)(1 + z 2 )(1 + z 3 ) · · · for, when these are multiplied out, each of the various terms which contribute to any particular power in the product is comprised of a product of distinct powers. Euler’s theorem in its analytic form is then just the identity 1 = (1 + z)(1 + z 2 )(1 + z 3 ) · · · (1 − z)(1 − z 3 )(1 − z 5 ) · · · throughout |z| < 1. To prove this identity, note that the product on the right can be expressed as ∞ " k=1

(1 + z k ) =

∞ " 1 − z 2k 1 − zk

k=1

(8)


283

But since the factors in the numerator, and those in the denominator, each determine a convergent infinite product, we can also express the above product as

lim N→∞ lim N→∞

N 3

(1 − z 2k )

k=1 N 3

lim N→∞ =

(1 − z k )

lim N→∞

k=1

N 3

(1 − z 2k )

k=1 2N 3

1

= lim (1 − z k )

N N→∞ 3

(1 − z 2k−1 )

k=1

k=1

which is precisely the expression on the left side of (8).

IV Another “Splitting” Problem Can we split the nonnegative integers into two sets A and B so that every nonnegative integer n can be expressed the same number of ways as the sum of two distinct members of A, as it can as the sum of two distinct members of B? If we experiment a bit and begin by placing 0 in A, then 1 ∈ B, otherwise 1 would be expressible as a + a but not as b + b . Similarly 2 ∈ B and 3 ∈ A (why?). Continuing in this manner we find that if the sets A and B exist, then A = {0, 3, 5, 6, 9, . . .} and B = {1, 2, 4, 7, 8, . . .}. But there is no obvious pattern, nor do we have any assurance that A and B even exist. So we turn to generating functions. Let

A(z) = z n and B(z) = zn n∈ A

n∈B

and let a(n) and b(n) denote the number of ways that n can be expressed as the sum of distinct members of A and B, respectively. (To be precise, we will assume that order does not count so that, for example, a(6) = b(6) = 1, rather than 2. Of course, if we counted order, it would have no effect on the problem of determining A and B.) Then the generating function for {a(n)} is 12 [ A2 (z) − A(z 2 )]. So if A and B exist, they must satisfy ( 1' ( 1' 2 A (z) − A(z 2 ) = B 2 (z) − B(z 2 ) 2 2 In addition, since A and B partition the integers, A(z) + B(z) =

1 1−z

(9)

(10)

Combining (9) and (10) shows that [ A(z) − B(z)] · or

1 = A(z 2 ) − B(z 2 ) 1−z

' ( A(z) − B(z) = (1 − z) A(z 2 ) − B(z 2 )

(11)

284


Now, (11) can be iterated, giving ' ( A(z 2 ) − B(z 2 ) = (1 − z 2 ) A(z 4 ) − B(z 4 ) and, applying (11) once again, ' ( A(z) − B(z) = (1 − z)(1 − z 2 ) A(z 4 ) − B(z 4 ) Continuing in this manner, we have A(z) − B(z) = (1 − z)(1 − z 2 ) · · · (1 − z 2

n−1

' ( n n ) A(z 2 ) − B(z 2 )

Finally, letting n → ∞, and using the facts that z n → 0, A(0) = 1, and B(0) = 0, we have ∞ " k A(z) − B(z) = (1 − z 2 ) (12) k=1

Expanding the above shows that A(z) − B(z) = 1 − z − z 2 + z 3 − z 4 + z 5 · · · =

∞

(−1)d(n) z n n=0

where d(n) is equal to the number of 1 s in the binary representation of the integer n. The above identity, together with (10), implies that if the sets A and B exist (and if we assume 0 ∈ A), then A consists of all nonnegative integers whose binary representations contain an even number of 1 s, while B consists of those integers whose binary form has an odd number of 1 s. So the solution, if it exists, is unique. But do the sets described above actually provide a solution to the original problem? In other words, if r A (n) and r B (n) denote the number of representations of n as the sum of two distinct integers of A and B, respectively, is r A (n) = r B (n) for all n? Fortunately, now that the generating functions have revealed the nature of the sets A and B, it is easy to see that the answer to the above question is YES! In fact, we can establish a 1 − 1 correspondence between sums of the form a + a and those of the form b +b , which equal the same number n, as follows. In the binary representations of a and of a , simply switch the 1 and the 0 in the first binary digit where they differ. For example, the number 12 is equal to the two sums 12 + 0 and 9 + 3, whose terms all belong to A. However we can also obtain representations of 12 of the form b + b by noting that 12 + 0 = (1100)2 + (0000)2 = (1000)2 + (0100)2 = 8 + 4, and 9 + 3 = (1001)2 + (0011)2 = (1011)2 + (0001)2 = 11 + 1. As an aside, it might be interesting to determine all integers for which r A (n) = r B (n) = 0. We leave it to the reader to verify that for n > 4, those are precisely the integers of the form 22k+1 − 1; that is, those integers whose binary representation consists of a single string of an odd number of 1 s.

19.5 An Analytic Proof of The Prime Number Theorem

285

19.5 An Analytic Proof of The Prime Number Theorem We conclude (this chapter and our textbook) with an analytic proof of the famous prime number theorem. 19.5 Prime Number Theorem Let π(N) denote the number of primes less than or equal to N. Then π(N) ∼ logN N . That is, π(N) log N → 1 as N → ∞. N Riemann seems to have been the first person to note the connection between the zeta function and the prime number theorem. The first complete proofs of the theorem, however, were not given until 1896, when de la Vallée Poussin and Hadamard each gave a complete but complicated “analytic” proof. The proof below is based on some key results concerning the zeta function along with an ingenious lemma of Chebychev and an application of Cauchy’s theorem. The properties of the zeta function which we will use are 3 (1) Euler’s Identity: ζ (z) = 1/ (1 − 1/ p z ) if Re z > 1 (See section 18.3) p prime

(2) (z − 1)ζ (z) is analytic and zero-free for Re z ≥ 1. (See theorems 18.9 and 18.10) According to (1), if Re z > 1 log ζ (z) = −

p prime

=

1 1 1 1 log(1 − z ) = + 2z + 3z + · · · p pz 2p 3p p prime

p prime,n≥1

1 npnz

Note, moreover, that

p prime,n≥2

is analytic for Re z >

1 2

L(z) =

1 1 1 = + 3z + · · · npnz 2 p 2z 3p p prime

(see Exercise 9). So the function

p prime

1 = log ζ (z) − pz

p prime,n≥2

is also analytic for Re z > 1. In addition, since log ζ (z) + log(z − 1)

1 npnz

(3)

286


is analytic for Re z ≥ 1 (see (2) above), so is the function L(z) + log(z − 1) as well as

− log p 1 1 = (4) L (z) + + z−1 pz z−1 p prime

Our proof will center on two functions: One is the analytic function φ(z) =

log p . pz

p prime

So, for future reference, note that, according to (4), 19.6 Lemma If φ(z) = p

prime

log p pz ,

then φ(z) −

1 z−1

is analytic for Re z ≥ 1.

The other function which plays a critical part in our proof is defined for positive real values x as θ (x) = p≤x log p. There are several equivalent versions of the prime number theorem. The one below, involving θ (x), is very straightforward. 19.7 Lemma The prime number theorem is equivalent to the assertion that θ (x) ∼ x. Proof Note that, on the one hand, θ (x) =

log p ≤

p≤x

log x = π(x) log x

p≤x

while, on the other hand, for any ε > 0, θ (x) ≥

x 1−ε < p≤x

log p ≥

' ( (1 − ε) log x = (1 − ε) log x π(x) − π(x 1−ε )

x 1−ε < p≤x

Since, obviously, π(x 1−ε ) ≤ x 1−ε , the two above inequalities combine to give π(x) log x 1 log x θ (x) θ (x) ≤ ≤ + ε x x 1−ε x x and the lemma follows by letting x → ∞.

Chebychev was well aware of the importance of θ (x) to the proof of the prime number theorem. While he was unable to prove that θ (x) ∼ x, he was able to prove


287

that, for sufficiently large x, 0.92 < θ(x) x < 1.11, which gave the corresponding double inequality for π(x)xlog x . We will not prove this double inequality, but we will use the following weaker result obtained by Chebychev.

19.8 Lemma (Chebychev) θ (x) < x log 16 Proof Note that

2n

is an integer whose prime factorization contains every prime number 2n 3 in the closed interval [n + 1, 2n] and hence p≤ . On the other hand, n n< p≤2n 2n 2n is only one of the terms in the binomial expansion of (1 + 1)2n = + n 0 2n 2n 2n + ··· + , so that ≤ 4n . Combining the two inequalities shows 1 3 2n n p ≤ 4n , and taking the logarithm of both sides: that n

n< p≤2n

log p ≤ n log 4.

(5)

n< p≤2n

Adding inequality (5) with n = 1, 2, 4, . . . , 2 M where M is the least integer for which 2 M+1 ≥ x, shows then that

log p < 2 M+1 log 4 < 2x log 4. θ (x) ≤ 1< p≤2 M+1

Note that, for Re z > 1, φ(z) =

log p p

pz

∞

= 1

dθ (x) =z xz

1

∞

θ (x) dx = z x z+1

∞

e−zt θ (et )dt

0

θ(x) since θ(x) x z = 0 for x = 1, and lim x→∞ x z = 0, according to Lemma 19.8. Hence ∞ φ(z + 1) = e−(z+1)t θ (et )dt (6) z+1 0 ∞ and subtracting the identity: 1z = 0 e−zt dt from (6) yields ∞ φ(z + 1) 1 − = e−zt [θ (et )e−t − 1]dt z+1 z 0

288


Let g(z) =

φ(z+1) 1 z+1 − z , and ∞ −zt f (t)dt 0 e

f (t) = θ (et )e−t − 1. Then

(i) g(z) = (ii) g(z) is analytic for Re z ≥ 0. This follows since Lemma 19.6 asserts that φ(z +1) has a simple pole with residue 1 at z = 0, and the same is obviously true for φ(z+1) z+1 . (iii) f (t) is bounded (according to Lemma 19.6) and locally integrable. According to the Analytic Theorem below, then, ∞ ∞ ∞ θ (x) − x f (t)dt = [θ (et )e−t − 1]dt = d x converges. x2 0 0 1

(7)

19.9 Analytic Theorem Let f (t), ∞t ≥ 0, be a bounded and locally integrable function and suppose ∞ that g(z) = 0 e−zt f (t)dt, Re z > 0, extends analytically to Re z ≥ 0. Then 0 f (t)dt exists (and equals g(0)). Proof

T For T > 0, set gT (z) = 0 e−zt f (t)dt. g is clearly an entire function. What we must show is that limT →∞ gT (0) = g(0). Let R be large and let C be the boundary of the region {z : |z| ≤ R, Re z ≥ −δ}, where δ = δ(R) > 0 is small enough so that g(z) is analytic inside and on C. Then g(0) − gT (0) =

1 2πi

(g(z) − gT (z))e zT (1 + C

z 2 dz ) R2 z

by Cauchy’s Theorem. On the semicircle C+ = C ∩ {z : Re z > 0} the integrand is bounded by 2B/R 2 , where B = maxt ≥0 | f (t)|, because

∞

|g(z) − gT (z)| = |

e

−zt

∞

f (t)dt| ≤ B

T

|e−zt |dt =

T

Be−T Re z Re z

while

2 Re z z2 z z¯ + z 2 |z + z¯ | = . | = | |= R R R2 R2 So, by the familiar M − L formula, the contribution to g(0) − gT (0) from the integral over C+ is bounded in absolute value by BR . For the integral over C− = C ∩ {z : Re z < 0}, we look at the terms involving g(z) and gT (z) separately. Since gT is entire, the path of integration for the term involving gT can be replaced by the semicircle C− = {z : |z| = R and Re z < 0}. The contribution to g(0) − gT (0) from the integral along this semicircle is also bounded by B/R, since |e zT | = e T Re z and |1 +

|gT (z)| = | 0

T

e−zt f (t)dt| ≤ B

T −∞

|e−zt |dt =

Be−T Re z , Re z < 0. |Re z|


289

Finally, the remaining integral over C− tends to 0 as T → ∞ because the integrand is the product of the function g(z)(1 + z 2 /R 2 )/z, which is independent of T, and the function e zT , which goes to 0 rapidly and uniformly on compact sets as T → ∞ in the half-plane Re z < 0. For example, we can choose T large enough so that −1 1 = {z ∈ C : Re z ≥ √ √1 < δ, and then split C− into two parts: C− } and − T

2 = {z ∈ C : Re z < C− −

−1 √ T

T

1 is bounded by M − L, where }. The integral over C−

M = Max|g(z)| for z ∈ C− √4 . T.

1 , which is bounded above by and L = the length of C−

bounded by

√ π RMe− T .

2 is The integral over C−

Hence lim sup |g(0) − gt (0)| ≤ 2B/R T →∞

and since R is arbitrary, the proof is complete. We now complete the proof of the Prime Number Theorem by showing that the convergence of the integral in (7) guarantees that θ (x) ∼ x. 19.10 Lemma Suppose h(x) is a non-decreasing function and ∞ h(x) − x dx x2 1

(8)

converges. Then h(x) ∼ x. Proof Assume that for some λ > 1, there are arbitrarily large values of x with h(x) ≥ λx. Then

λx x

h(t) − t dt ≥ t2

λx x

λx − t dt = t2

λ 1

λ−t dt > 0 t2

for all such x, contradicting (8). Similarly, the inequality h(x) ≤ λx with λ < 1 would imply

x

λx

h(t) − t dt ≤ t2

λx x

λx − t dt = t2

1

λ

λ−t dt < 0 t2

which, according to (8), cannot be true for a fixed λ and arbitrarily large values of x.

290


Exercises 1. Show that tan z = z has only real solutions. 2. With C N as in 19.1, use the fact that f 2 (z) =

z2 (1 + z 2 )(tan z − z)

∼−

1 z

to conclude C f 2 (z) dz → −2π i. (Compare with 12.1, Example 3.) Conclude that N Var

sin2 x x2

= e2 − 5

by considering the various contributions to lim

N →∞ C N

f 2 (z) dz.

3. Recall (Section 16.2) that ez = z has infinitely many solutions z k ; k = 1, 2, . . .. Find

∞

2 k=1 1/z k .

4. Find all solutions to the system of equations in 19.3 if we require only ak , bk ≥ 0 for all k. 5. Show that the positive integers cannot be partitioned into a finite number of arithmetic progressions if one of the differences is relatively prime to the others. 6.* (The following exercise shows how generating functions can be used to "solve a difference equation"; i.e., to find an explicit form for a recursively-defined sequence.) Suppose is defined a sequence n. c z recursively by c0 = c1 = 1 and cn+2 = cn+1 + 2cn for n ≥ 0. Let F(z) = ∞ n 0 a. Prove by induction that cn ≤ 3n , and hence show that the radius of convergence of F(z) ≥ 13 . b. Show that (1 − z − 2z 2 )F(z) = 1, and express F(z) as a sum of two simple rational functions. c. Find (a closed form) for the sequence of coefficients {cn } ∞ 2 n n 7.* Let cn = 12 + 22 + · · · + n 2 , and F(z) = ∞ 1 cn z . Show that (1 − z)F(z) = 1 n z and thereby obtain a closed form for cn . 8.* Use the method outlined in 19.4 II to find a formula for C(n), the number of ways of expressing the positive integer n as a nonnegative integral combination of 3 s and 4 s. 9. Prove

p prime n≥2

1 npnz

is analytic in Re z > 12 . [Hint: Show

1 npnz
5/2; a region. d. The upper half of the unit disc, centered at the origin; a region. e. The unit circle centered at the origin; not a region. f. The circle centered at 12 i with radius 12 ; not a region.

291

292

Answers

g. The set of points satisfying r 2 < 2 cos 2θ, r = 0; not connected and, therefore, not a region. 17. Arg z−1 z+1 = Arg(z − 1) − Arg(z + 1).Consider the diagram below for Re z > 0 and an analogous argument for Re z < 0.

z

α

β –1

1

21. a. For |z| ≤ r < 1, |kzk | < kr k < ∞. Apply Theorem 1.9. 1 1 . Apply Theorem 1.9. b. For Re z ≥ 0, k 2 +z ≤ k2 22. Assume S is polygonally connected, and S is “disconnected” by open sets A and B with a ∈ A, b ∈ B. Consider the polygonal line L(t), 0 ≤ t ≤ 1, connecting a and b, and let c = L(t0 ), t0 = sup{t1 : L(t) ∈ A, 0 ≤ t ≤ t1 }. Then, c ∈ A, but, because A is open, so is L(t0 + δ), unless c = b. 23. Note that no curve in S (of finite length) can connect a point of S, not on the y axis, with a point on the y axis. Nevertheless, S cannot be “disconnected” because any open set which contains a point of the form (0, t), −1 < t < 1, would also have to contain points of the form y = sin 1x , x > 0. ζ0 24. ζ ≥ ζ0 ⇒ x 2 + y 2 ≥ 1−ζ 0 ζ 1 + Bξ 1−ζ + 25. A(x 2 + y 2 ) + Bx + C y + D = 0 is equivalent to A 1−ζ 1 Cη 1−ζ + D = 0 or Aζ + Bξ + Cη + D(1 − ζ ) = 0, ζ = 1. Consider A = 0 and A = 0. a 26. If P(z) = an z n + an−1 z n−1 + · · · + a0 , an = 0, then P(z) = z n an + n−1 z + a0 · · · + z n , etc. iy 1 x = x 2 +y 27. a. Use formulas (3) for (ξ, η, ζ ) and the fact that 1z = x+iy 2 − x 2 +y 2 to show that (ξ , η , ζ ) = (ξ, −η, 1 − ζ ). b. Consider the corresponding points on and apply 27 (a). 28. Note that according to (27), Aξ +Bη+Cζ = D is equivalent to Aξ −Bη −Cζ = D − C.

Answers

293

Chapter 2 1. For P(x, y) = (x + i y)n , Px = n(x + i y)n−1 and Py = i n(x + i y)n−1 . 3. a. Analytic: P(z) = z 3 − z. b. Not analytic. c. Analytic: P(z) = −i z 2 . 4. Py = i Px would imply Py ≡ Px ≡ 0 because both Py and Px would be imaginary. 5. In (3a), P (z) = 3z 2 − 1 = 3x 2 − 3y 2 − 1 + i 6x y. In (3c), P (z) = −2i z = 2y − 2i x y. 6. Use the usual properties of limits, etc. n n 7. limh→0 (z+h)h −z = limh→0 [(z + h)n−1 + (z + h)n−2 z + · · · + z n−1 ] = nz n−1 8. log Sn = logn n → 0 as n → ∞, so Sn = elog Sn → 1. 9. a. 1. b. 1/2. 10. a. R. b. R. c. R 2 . 11. The radius of convergence must be greater than or equal to min(R1 , R2 ). It may exceed both if, e.g., b n = −an for all n and R1 , R2 < ∞. cos nθ sin nθ 12. Use the facts that converges for all θ and converges for n n θ : 0 < θ < 2π (both can be proven by Dirichlet’s Test). n 13. a. Write an = a1 aa21 · · · aan−1 and note that, according to the hypothesis, aj for any > 0, there is some k such that j > k ⇒ a j −1 − L < . Thus, Mk (L − )n−k < an < Mk (L + )n−k , etc. 1 b. Note that, for an = n1! , an+1 an = n+1 and apply (13a). 14. a. ∞. b. ∞. c. e. d. ∞. k 17. If |ak | and |bk | converge and if dk = clearly, j =0 |a j ||bk− j |, then, n n ∞ ∞ ∞ d ≤ |a | · |b | ≤ |a | |b | so that d k conk=0 k k=0 k k=0 k k=0 k k=0 n k verges. Moreover An Bn = Cn + Rn where |Rn | = a b ≤ d n+1 + k, j =0 k j n 0 and let ϕ2 : R → U with ϕ2 (z 0 ) = 0, ϕ2 (z 0 ) = M ∗ ≥ M. Let f (z) = ϕ2 o ϕ1−1 (z). Then f is analytic in the unit disc; | f | < 1 there, f (0) = 0 and f (0) = M∗

ϕ2 (z 0 )

ϕ1 (z 0 )

=

M∗ M

≥ 1. Hence, by Schwarz’ lemma,

= M and f (z) = z, i.e., ϕ2 = ϕ1 .

Chapter 15 2. eiz z 3. e z maps the lines y = ± π2 onto the imaginary axis; hence, ee 1 on the z boundary of D. If f (z) A ee in D, then g(z) = f (log z) A e|z| in the right half-plane. 2

Answers

313

Chapter 16 2 1. u + ν = Re[(1 − i ) f ]; uv = Re −i2f . 2. If g = f x , then g x x + g yy = ( f x x + f yy )x = 0. Similarly, f y is harmonic. 3. If g = u 2 , gx x + g yy = 2u(u x x + u yy ) + 2(u 2x + u 2y ) = 2(u 2x + u 2y ) which cannot be identically zero unless u is constant. 4. A direct calculation shows that, if u = log(x 2 + y 2 ), u x x + u yy = 0. If u were the real part of an analytic f (z), z = 0, f would have to agree up to an additive constant with the analytic function log z in the simply connected domain 0 < Arg z < 2π. But then, f (z), like log z, would not be continuous on the positive real axis. 5. Note that, if v(r, θ ) = u(r cos θ, r sin θ ), then, r 2vrr +r vr +v θθ = r 2 (u x x +u yy ). Hence, Laplace’s equation becomes r 2 νrr + r νr + νθθ = 0 or r

∂ (r vr ) + v θθ = 0. ∂r

∂ i.e., ∂r (r vr ) + 1r v θθ = 0. If v depends on r alone, v θθ = 0, and the above differential equation implies v = a ln r + b.

Note also that, if vr = 0 and v is harmonic, then v θθ = 0, or v(θ ) = aθ + b. (ζ ) (ζ ) (z−¯z ) f (ζ ) 6. 2πi f (z) = C R ζf −z dζ − C R ζf −¯ z dζ = C R (ζ −z)(ζ −¯z ) dζ . Let R be the semi z ) f (ζ ) circular arc |z| = R, Im z ≥ 0. Then, as R → ∞, R (ζ(z−¯ −z)(ζ −¯z ) dζ → 0 by the M-L formula, so that ∞ y f (t) 2πi f (z) = 2i dt, (t − x)2 + y 2 −∞ and, dividing by 2πi and equating real parts, 1 ∞ yu(t) u(x + i y) = dt π −∞ (t − x)2 + y 2 7. Note that Re(z 3 ) = x 3 − 3x y 2 = 4x 3 − 3x on the unit circle. Hence u(x, y) = 8. u(z) =

3 2

− π1 Arg

z−1 z+1

1 3 (x − 3x y 2 + 3x). 4

where the Arg takes values between

z−1 u(z) = k ⇒ Arg z+1

=θ =π

π 2

3 −k . 2

and

3π 2 .

314

Answers

Thus, for 12 < k < 1, z lies on the upper arc of a circle for which [ − 1, 1] is a chord and for which the lower arc has 2θ degrees. For 0 < k < 12 , z lies on congruent arcs in the lower half-plane. The chord [ − 1, 1] is the level curve for k = 12 . 9. π1 Arg z = π1 θ (see the note following Exercise 5). 10. sin z maps the strip onto the upper half-plane with the boundary of the strip being mapped onto the real line segments (−∞, −1), [ − 1, 1], (1, ∞). In the upper half-plane: w > 0, Arg(w2 − 1) has the values 2π, π, 0 on the intervals (−∞, 1), (−1, 1) and (1, ∞), respectively, and Arg(w2 − 1) = Im[ log(w2 − 1)] is the imaginary part of a function analytic in the upper halfplane. Thus, the desired solution is u(x, y) =

1 1 Arg(sin2 z − 1) = Arg(− cos2 z) π π

Note, for example, that on the y axis, u(0, y) = π1 Arg(− cosh2 y) = 1. 11. By Theorem 16.3, if e z − P(z) does not have Infinitely many zeroes, e z − P(z) = Q(z)e R(z) where Q, R are polynomials. Considering the growth at infinity, it follows that R(z) = z, Q(z) = 1, and P(z) = 0. Similarly, for sin z − P(z). 12. If a function f of order j does not have infinitely many zeroes, f (z) = Q(z)e P(z) . But if f = 0, Q is a constant and f can be written in the form f (z) = e P(z) . Finally, because f is of order j, P is a polynomial of degree j .

Chapter 17 1. 2.

3N k=2

3N

k=2

'

1−

1+

3 N (k−1)(k+1) 1 = k=2 = N+1 . Hence PN → 12 as N 2 k2 ( k 2N ' ( k N (−1) = 32 23 54 45 · · · 1 + (−1) . Hence, k N PN =

and PN → 1 as N → ∞.

⎧ ⎨1 ⎩1 + 1 N

→ ∞.

if N is odd if N is even

z k2 z k3 2 − + | ≤ 12 . Hence, if |z k |2 4. log(1 + z k ) − z k = 3 − + · · · z k if |z k 2 converges, so does [ log(1 + z k ) − z k ] and, because z k converges, it follows that log(1 + z k ) converges. By Proposition 17.2, then, (1 + z k ) converges.

Answers

315

converges, the convergence of log(1 + z k ) is equiva z2 z3 lent to the convergence of [ log(1 + z k ) − z k ]. But the latter is k − 2k + 3k − + · · · and, for k ≥ 4, log(1 + z k ) − z k ≤ −1 log(1 + z k ) diverges. 6k so that N−1 N −1 1 2 2 2 2 k ) = 1 + z + z + ···+ z → ∞ 6. (1 + z)(1 + z ) · · · (1 + z k=0 z = 1−z uniformly for |z| ≤ r < 1. 3∞ z 7. k=1 1 − k 2 5. Because

zk =

(−1)k √ k

8. Using the power series expansion for sin z, it can be seen that

√ sin π z √ π z

is entire

and equal to zero if z = k 2 ; k = 1, 2, · · · . Note, also, that, according to Proposition 17.8, the solutions in (7) and (8) are identical. 3∞ 2 4z cos π z is entire and zero-free. As in 9. f (z) = k=1 1 − (2k+1)2

10.

11. 12.

13.

Proposition 17.8, it can be shown (considering the magnitude of f on a square of side 2N centered at the origin) that | f (z)| ≤ A exp(|z|3/2 ) and that f is, in fact, constant, so that f (z) = f (0) = 1. The product form also be derived from the identity: sin 2π z cos π z = . 2 sin π z ' ( 1 3 1 a. ∞ 1 − k=1 k(1−z) exp k(1−z) . b. Le {z k } be a sequence of distinct points with limk→∞ z k = z 0 . Then, an 1 entire function can be defined with zeroes at the points λk = z0 −z . Setting k g(z) = f z01−z , g will be analytic for z = z 0 and equal to zero at the points of the original sequence {z k }. b ϕ(ζ,t ) b 1 ϕ(z,t ) b ∂ 1 F (z) = 2πi C a (ζ −z)2 dt dζ = a 2πi C (ζ −z)2 dζ dt = a ∂z (ϕ(z, t))dt. x− h(u)y Because h is continuous, |h| ≤ M on [α, β]. For any > 0, α (u−x) 2 +y 2 du β h(u)y M y(β−α) and x+ (u−x)2+y 2 du are each bounded in absolute value by whereas 2 x+ h(u)y −1 where x − < x < x + . Hence, as x− (u−x)2 +y 2 du = h(x) · 2 tan y β h(u)y y → 0, α (u−x)2 +y 2 du can be made arbitrarily close to πh(x). 1 dt f (z) = 0 1−zt = − log(1−z) which is analytic in C − [1, ∞). By the argument z principle, Arg(1 − z) = 2πi as z circles the point z = 1; hence, f has a jump discontinuity of 2πi to the lower halfx as z crosses from the upper half-plane ∞ dt plane at z = x > 1. Note also that, if we consider g(z) = 0 et −z (Example 2 1 dt . following 17.9), setting u = e−t , it can be shown that g(z) = 0 1−zt

316

Answers

Chapter 18 1. Assuming log z is the principle branch, i.e., Im log z = 0 on the positive axis, it follows that Im g1 (z) will be between −π and − π2 in the third quadrant, whereas 3π Im g2 (z) will be between πn and 2 . 3. Note that f (−z) = an z ; an ≥ 0, and apply Theorem 18.3. ∞ −nt −2/3 1 1 4. a. Because n1/3 = (1/3) t dt, 0 e ∞

zn 1 n = (ze−t ) t −2/3 dt (1/3) 0 n 1/3 ∞ 1 z dt = 2/3 (1/3) 0 t (et − z) which is analytic outside of the interval [1, ∞) b. Since

1 n 2 +1

=

∞ 0

e−nt sin t dt,

∞ zn n = (ze−t ) sin t dt n2 + 1 0 ∞

n = (ze−t ) sin t dt = 0

0

∞

et sin t dt et − z

which is analytic outside of the interval [1, ∞) 5. Make the change-of-variables u = nt. 6. Setting u = t 2 yields √ ∞ 1 π 1 −t 2 = . e dt = 2 2 2 0 t2 −t /n − 1 − 7. Because e−t /n = 1 − nt + 2n 2 − +··· , 0 ≤ e −t 2 n so that e−t − 1 − nt ≤ e · e 2nt and

n 0

t n

≤

t2 , 2n 2

if t ≤ n,

n n t n e t z−1 1 − dt − t z−1 e−t dt t z−1 e−t t 2 dt n 2n 0 0 e ≤ (Re z + 2) 2n

which approaches 0 as n → ∞. 9. f (z) = 1 − 21z + 31z − + · · · = 1 −

2 2z

ζ (z) so that f is certainly analytic, like

−2 ζ (z), for z = 1. Moreover, limz→1 f (z) = limz→1 2z2(z−1) = ln 2 so that f is analytic at z = 1 as well. 1 10. Because ζ (z) → ∞ as z → 1, 1 − 1p diverges to 0. Because conp2 1 verges, this implies that p diverges (see Chapter 17, Exercise 4). z

Answers

317

Chapter 19 1. Consider f (z) = tan z − z inside the square centered at the origin and with sides of length 2π N, whose boundary is denoted C N . Then, the number of poles of f (z) inside C N = the number of zeros of cos z inside C N = 2N. The number of real zeros of f inside C N is 2N +1 since f has a triple zero at the origin and tan x = x has exactly one solution in each of the intervals [(2k − 1) π2 , (2k + 1) π2 ]; k = ±1, ±2, . . . , ±(N − 1). Let c = the number of complex, nonreal zeros of tan z − z inside C N . Then 1 2πi

CN

tan2 z dz = Z − P = 1 + c tan z − z

by the calculations above. By the usual M − L estimates and the fact that | tan z| < 1 + throughout C N (where → 0 as N → ∞), it follows that Z − P = 1 + c < 2. Hence c = 0. 2. With f 2 (z) =

z2 , (1+z 2 )(tan z−z)

note that

CN

f2 (z)dz → −2πi whereas

⎡

CN

⎢ f 2 (z)dz → 2πi ⎢ ⎣

⎤ ∞

sin2 x k x k2

k =1 xk = 0

⎥ + Res( f 2 ; i ) + Res( f 2 ; −i ) + Res( f 2 ; 0)⎥ ⎦.

Note, then, that Res( f 2 ; i ) = Res( f 2 ; −i ) = −

e2 +1 4

z2

whereas Res( f 2 ; 0) = 3 3

because tan z−z has a simple pole at z = 0 and limz→0 tanzz−z = 3. Hence, 2 ∞ sin2 xk sin2 x k e2 −7 sin x =2 ∞ = and Var + 2 = e2 − 5. 2 2 k=0 k=1 2 xk x k =0

xk

e z −1

x k =0

xk

3. Let f (z) = z 2 (ez −z) . Then C N f (z)dz → 0 as N → ∞ if C N is a square centered at the origin with sides of length 2π N. At the same time, C N f (z)dz → 1 + Res( f ; 0) where the sum is taken over the zeros of e z −z. Because 2πi z k2 1 Res( f ; 0) = +1 it follows that 2 = −1. zk

4. As in Section 19.3, a solution {ak }, {bk }, would imply 1 + a1 z +

a2 z 2 α2 z 2 + · · · = eαz = 1 + αz + + ··· 2! 2!

1 + b1 z +

b2 z 2 β 2z2 + · · · = eβz = 1 + βz + + ··· 2! 2!

318

Answers

so that α = a1 and ak = a1k , k = 2, 3, . . ., and β = b1 , bk = b1k ; k = 2, 3, . . .. Thus, there would be infinitely many solutions of the form {ak }, { bk } with a1 , b1 ≥ 0; a1 + b1 = 2 and ak = a1k ; bk = b1k for k = 2, 3, . . .. 5. Suppose d1 is relatively prime to all d j , j = 1, and assume that the desired partition is possible. Then, as in Section 19.5, z 2a z a1 z ak z = + + · · · + 1−z 1 − z d1 1 − z d2 1 − z dk for |z| < 1. Then, if we let z → e2πi/d1 , the first term on the right side of the equality would approach infinity whereas all the others would approach a finite limit. Thus, the partition is impossible. (In fact, according to this argument, the partition would be impossible as long as one of the differences is not a divisor of any of the others.) ∞ 1 ∞ px 1 ∞ 1 2 1 1 1 9. n=2 np nz = n=2 np nx ≤ 2 n=2 p nx = 2 p 2x ( p x −1) ≤ p 2x for x > 2 . ∞ 1 ∞ 2 2 In fact, for x ≥ 12 + δ, p prime p 1+2δ ≤ n=2 np nz ≤ n=1 n 1+2δ < ∞. Thus,

p prime

1 n ≥ 2 np nz

is uniformly convergent on compacta and is analytic in the

p prime

half-plane Re z >

1 2.

References

1. Ahlfors, L. Complex Analysis, third edition, McGraw-Hill, NY, 1979 2. Apostol, T. Introduction to Analytic Number Theory, Springer-Verlag, NY, 1976 3. Bak, J., Ding, P. and Newman, D.J. Extremal Points, Critical Points, and Saddle Points of Analytic Functions, American Mathematical Monthly 114 (2007), 540–546 4. Bak, J. and Ding, P. Shape Distortion by Analytic Functions, American Mathematical Monthly 116 (2009), 143–150 5. Conway, J. Functions of One Complex Variable, 2nd edition, Springer-Verlag, 1978 6. Carathéodory, C. Theory of Functions of a Complex Variable, Chelsea Publishing Company, NY, 1954 7. Davis, P. The Schwarz Function and Its Applications, The Mathematical Association of America, 1974 8. Henrici, P. Applied and Computational Complex Analysis, Wiley, 1986 9. Lang, S. Complex Analysis, 2nd edition, Springer-Verlag, NY 1985 10. Markushevich, A.I. Theory of Functions of a Complex Variable, (English Edition) PrenticeHall, Englewood Cliffs, NJ, 1965. 11. Milne-Thomson, L.M. Theoretical Hydrodynamics, Macmillan, NY, 1968 √ 12. Nahin, P. An Imaginary Tale: The Story of −1, Princeton University Press, Princeton, NJ 1998 13. Needham, T. Visual Complex Analysis, Oxford University Press, 1997 14. Nehari, Z. Conformal Mapping, McGraw-Hill, 1952 15. Newman, M.H.A. Elements of the Topology of Plane Sets of Points, Cambridge University Press, 1964 16. Newman, D.J. Simple Analytic Proof of the Prime Number Theorem, American Mathematical Monthly 87 (1980), pp. 693–696 17. Pólya G. and G. Szeg˝o, G. Problems and Theorems in Analysis I, Springer-Verlag, 1972 18. Stillwell, J.C. Mathematics and Its History, Springer-Verlag, 1989 19. Titchmarsh, E.C. The Theory of Functions, 2nd edition, Oxford University Press, London, 1939. 20. Wilf, H. Generatingfunctionology, Academic Press, 1994 21. Zagier, D. Newman’s Short Proof of the Prime Number Theorem, American Mathematical Monthly 104 (1993), 705–708. 22. Zalcman, L. Picard’s Theorem without Tears, American Mathematical Monthly, 85 (1978), 265–268

319

Appendices

I A Note on Simply Connected Regions The definition of simple connectedness (8.1) led to a relatively easy proof of the General Closed Curve Theorem (8.6). At the same time, while Definition 8.1 was somewhat complicated, we were able to establish the very intuitive result that a simply connected region contains, along with any closed polygonal path, all of the points which are “inside” the path. (See Lemma 8.3 and Exercise 6 of Chapter 8). This property of a simply connected region can be generalized. That is, a simply connected region contains, along with any closed curve, all the points inside the curve. The difficulty in proving the general result lies in defining the “inside” of a general closed curve. If we limit ourselves to smooth closed curves, however, we can use complex integrals to define the “inside” of the curve and we can prove the above property of simply connected regions. Definition

dz If is a smooth closed curve, we say that a point z 0 ∈ is inside if z−z = 0. 0 The totality of such points is called the inside of . Note that a similar definition (10.4), under more limited circumstances, is given in Chapter 10. Lemma ˜ If D is a simply connected region, is a closed curve contained in D and z 0 ∈ D, then there exists a differentiable curve γ(t) which connects z 0 to ∞ and which does not intersect . Proof According to Definition 8.1, there exists a continuous curve γ , connecting z 0 to ∞ ˜ < . If we take = 1 d(, D), ˜ γ will not intersect . Moreover, since with d(γ , D) 2 γ → ∞, for some N, t ≥ N ⇒ |γ(t)| ≥ max{|z| : z ∈ }. We can, then, redefine γ(t) = Nt γ (N) for t ≥ N so that γ will be differentiable (γ will actually be constant) for t ≥ N. Finally, because γ(t), 0 ≤ t ≤ N, can be uniformly approximated by a differentiable curve, there exists a curve γ with all of the desired properties.

321

322

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Theorem If D is a simply connected region and is a smooth closed curve contained in D, then the inside of is contained in D. Proof

dz If not, there would be z 0 ∈ D˜ for which z−z = 0. Let γ be a differentiable curve, 0 connecting z 0 to ∞ and not intersecting (as in the above lemma), and define dz , t ≥ 0. I (t) = z − γ (t) I (t) can be differentiated with respect to t and dz I (t) = γ (t) . [z − γ (t)]2 The above integral is clearly 0 (for all t) since the integrand has, as a primitive, −1 the function z−γ (t ) . Thus we can conclude that I (t) is constant. On the other hand, I (0) = 0 (since γ(0) = z 0 ) and I (t) → 0 as t → ∞ since the integrand approaches 0 uniformly, which yields the desired contradiction.

II Circulation and Flux as Contour Integrals Let C be a closed curve given by z(t) = x(t) + i y(t). Then a vector tangent to C is given by dx dy z˙ (t) = +i dt dt and a normal vector to C is given by dx dy −i . dt dt (If C is parametrized so that the tangent points in the counter-clockwise direction, the above normal vector points “outward.”) Suppose g = u + i v represents a flow function throughout C. Then the circulation around C is found by integrating the tangential component of g against the arclength, and the flux across C is given by the corresponding integral of the normal component of g. Let σ, τ represent the circulation and flux, respectively, and recall that the component of a vector α in the β|). Then direction of β is given by ( α ◦ β/| dy dx σ = +v dt = u u dx + v dy dt dt C C and τ=

dx dy −v dt = u u d y − v d x. dt dt C C

Appendices

323

Note, finally, that if f = g¯ = u − i v, f (z)dz = (u − i v)(d x + i d y) = σ + i τ. C

C

III Steady-State Temperatures; The Heat Equation Let the function u denote the temperature at the points of a region D and assume that u is independent of time. Then u = u(x, y) is a real-valued function of the position (x, y), and we wish to show that it is harmonic. To this end, we note two basic facts: 1. Heat flows in the direction of cooler temperatures, and the amount of heat crossing a curve per unit of time is proportional to the length of the curve and the difference in temperature across the two sides. Thus the amount of heat crossing a horizontal line of length x is equal to K u y x, while the amount of heat crossing a vertical line of length y is given by K u x y. 2. The total increase in heat (the amount of heat entering minus the amount of heat leaving) in any square S ⊂ D must be zero. Otherwise, the temperature at points of S would change, contrary to our assumption that u is independent of time. Using these two facts, we can obtain the following proof that u is harmonic, assuming u ∈ C 2 . Suppose that S is any square in D with horizontal and vertical sides of length h and assume without loss of generality that the lower left vertex is (0,0). Note that for any function f (x, y) with continuous partial derivatives at the origin, f (x, y) − f (0, 0) = f (x, y) − f (x, 0) + f (x, 0) − f (0, 0) = y f y (x, ξ ) + x f x (η, 0) so that (3)

f (x, y) − f (0, 0) = y( f y (0, 0) + 1 ) + x( f x (0, 0) + 2 )

where 1 and 2 → 0 as (x, y) → (0, 0). To obtain a formula for the change in the amount of heat in S per unit time, we first calculate the loss of heat through the top side minus the increase through the bottom side. According to (1), over any subinterval x, this is given by [K u y (x, h) − K u y (x, 0)]x. But according to (3), u y (x, h) = u y (0, 0) + xu yx (0, 0) + hu yy (0, 0) + 1 x + 2 h u y (x, 0) = u y (0, 0) + xu yx (0, 0) + 3 x so that

u y (x, h) − u y (x, 0) = hu yy (0, 0) + 4 h

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where 4 → 0 as h → 0. The net decrease in heat from the (two) subintervals thus equals K [hu yy (0, 0) + 4 h]x, and the net loss through the top and bottom sides is given by K [h 2 u yy (0, 0) + 4 h 2 ]. Similarly, the net loss through the vertical sides is given by K [h 2 u x x (0, 0) + 5 h 2 ]. and since the overall decrease must be zero, u x x (0, 0) + u yy (0, 0) + 4 + 5 = 0. Since, finally, h could have been chosen as small as possible, we conclude u x x (0, 0) + u yy (0, 0) = 0 and since the origin is in no way special, it follows that u is harmonic throughout D.

Index

Absolute value, 5 Absolutely convergent product, 245 sum, 13 Analytic arc, 102, 206 branch of log z, 113 continuation, 257, 263 function, 38 part, 124 polynomial, 21, 23 Angle between curves, 169 Annulus, 120, 232 Arc length, 50 Argand, J., 4 Argument, 6 Principle, 136 Associative law, 2 Automorphism, 182 Axis imaginary, 4 real, 4 Barrier, 197 Bilinear transformation, 177 Binomial coefficients, 154 Boundary, 13 natural, 259 Bounded set, 13 C-analytic, 86 C-harmonic, 229 Canonical regions, 200 Carathéodory proof of open mapping theorem, 93 proof of Rouche’s theorem, 142 Carathéodory-Osgood Theorem, 205

Cardan, J., 1, 11 Casorati-Weierstrass Theorem, 119 Cauchy product, 28, 33 Residue Theorem, 133 sequence, 12 Cauchy Integral Formula for analytic functions, 79 for entire functions, 61, 74 general, 138 Cauchy-Riemann Equations, 24, 35, 36 Chebychev, 286, 287 Circle of convergence, 27, 257 Circulation, 322 Closed curve, 52 Closed Curve Theorem for analytic functions, 78 for entire functions, 56 general, 112 Closed set, 13 Closure, 13 Commutative law, 2 Compact, 14 Complement, 13 Complex integral, 45 number, 1 plane, 4 Conformal equivalence, 169, 172, 175 mapping, 169, 175 Conjugate, 5 Connected set, 14 Continuous function, 14 Convergence absolute, 13, 243 circle of, 27, 257

325

326 of Newton’s method, 71 quadratic, 72 radius of, 26 uniform, 15, 99 Convergent integral, 143, 144, 147 product, 241 sequence, 12 Convex, 67 Critical point, 87 Cross-ratio, 185 Cubic equation, 9 Curve closed, 52 Jordan, 132 level, 238 piecewise differentiable, 45 regular closed, 133 simple closed, 52 smooth, 45 Curves smoothly equivalent, 46 Cyclotomic equation, 19 Define Integral complex, 45 real, 143 Deleted neighborhood, 117 Dense set, 119 Derivative complex, 24, 35 partial, 23 Descartes, R., 1 Differentiable function, 24 Differential equation, 222 Dirichlet Problem, 229 Dirichlet series, 251 Disconnected, 14 Distributive law, 2 Entire function, 38 Equicontinuous set of functions, 202 Essential singularity, 118 Euler constant, 268 theorem, 282 Euler, L., 1 Exponential function, 40 Extended plane, 18 Field of complex numbers, 1 Fixed point, 70, 184 Fluid flow locally irrotational and source-free, 195

Index totally irrotational and source-free, 197 Flux, 195, 322 Fourier Uniqueness Theorem, 275 Fraction, partial, 125 Function analytic, 38 complex continuous, 14 differentiable, 24 entire, 38 even, 74 exponential, 40 Gamma, 265 harmonic, 225 inverse, 38, 174 locally 1-1, 170 logarithm, 113 meromorphic, 135 of z, 21 rational, 125 trigonometric, 41 Zeta, 257, 268 Functions, equicontinuous, 202 Fundamental Theorem of Algebra, 66, 91 Fundamental Theorem of Calculus, 51 Gamma Function, 265 Gauss, C., 1 Gauss-Lucas theorem, 67 Generating functions, 273, 277, 278 Hadamard, viii Hamilton, W., 1 Harmonic function, 225 Heat Equation, 232, 323 Hurwitz’s Theorem, 139 Imaginary axis, 4 part, 5 Infinite product, 241 Infinity, point at, 17 Integral, 147 complex definite, 45 line, 46 real definite, 143 Integral Theorem for analytic functions, 78, 111 for entire functions, 54 Inverse function, 38 Isolated singularity, 117 Isomorphism, 2 Jordan Curve Theorem, 132 Jordan region, 205

Index Kelvin’s Theorem, 199 Kernel function Poisson, 228 Landau proof of the maximum modulus theorem, 91 theorem, 263 Laplace Equation, 232 transform, 262 Laurent expansion, 120 analytic part, 124 principal part, 124 Laurent series, 120 Level curve, 238 Levels of polygonal path, 109 Lim sup, 26 Liouville’s Theorem, 65, 96 Extended, 66 for Re f , 234 Locally 1-1, 170 Logarithm function, 113 M-L Formula, 49 M-Test, 15 Mapping bilinear, 177 conformal, 169 Theorem, 200 Maximum Modulus Theorem for analytic functions, 86 for harmonic functions, 227 generalized, 215 Mean-Value Theorem for analytic functions, 85 for harmonic functions, 226 Meromorphic function, 135 Minimum Modulus Theorem, 87 Modulus, 5 Moments, 261 Morera’s Theorem, 98 Neighborhood, 13 deleted, 117 Newton basins, 73 Newton’s method, 68 Open Mapping Theorem, 93 Order in the complex plane, 4 of a pole, 118 of a zero, 67 Order of an entire function, 236, 239

327 Partial derivative, 23 Partial fraction decomposition, 125 Partition problem, 278 Phragmén-Lindelöf Theorem, 218 Picard’s Theorem, 120 Piecewise differentiable curve, 45 Plane complex, 4 extended, 18 Point at infinity, 16 fixed, 70, 184 regular, 258 saddle, 89 Poisson Integral Formula for a disc, 229 for a half-plane, 229 Poisson Kernel, 228 Polar coordinates, 8 Pole, 118 Polygonal line, 14 Polygonal path, 16 levels of, 109 Polygonally connected, 14 Polynomial analytic, 21 real, 18, 74 Power series, 25, 28, 80 Prime number theorem, 285 Principal parts, 124 Quadratic equation, 3 Radius of convergence, 26 Rational function, 125 Real axis, 4 part, 5, 225, 226 polynomial, 18, 74 Rectangle Theorem for analytic functions, 77 for entire functions, 52, 59 Reflection Principle, 101 Region, 14 convex, 116 simply connected, 107 Regular closed curve, 133 Regular point, 258 Removable singularity, 117 Residue, 129 Residue Theorem, 133 Riemann Mapping Theorem, 200 Principle of Removable Singularities, 118

328 sphere, 17 Rotation, 175 Rouché’s Theorem, 137, 142 Saddle point, 89 Schwarz Lemma, 94 Reflection Principle, 101 Schwarz-Christoffel transformation, 187 Sequence Cauchy, 12 Convergent, 12 Series, 12 Dirichlet, 251 power, 26, 28, 80 Simply connected, 107, 321 Singularity, 258 essential, 118 isolated, 117 removable, 117 Sphere Riemann, 16 Square root, 3 Stereographic projection, 16 Streamlines, 213

Index Tangent to a curve, 169, 322 Taylor Expansion for an analytic function, 80 for an entire function, 63 Triangle inequality, 5, 19 Uniform convergence on compacta, 99 Uniqueness Theorem for analytic functions, 83 for power series, 31 Value absolute, 5 Variation, 273 Vector sum, 4 Velocity vector, 196 Wallis, J. , 4 Weierstrass Product Theorem, 244 Winding number, 130 Zero of multiplicity k, 67, 75 Zeroes of a real polynomial, 18 Zeroes of entire functions, 236, 244 Zeta function, 257, 268

Complex Analysis, Third Edition (Undergraduate Texts in Mathematics)

Undergraduate Analysis (Undergraduate Texts in Mathematics)

Mathematical Logic: Undergraduate Texts in Mathematics (Undergraduate texts in mathematics)

Linear Algebra, Third Edition (Undergraduate Texts in Mathematics)